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Computational OT #2 - Entropic regularization

Gabriel Peyré
April 09, 2025
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Computational OT #2 - Entropic regularization

A course on computational optimal transport

Gabriel Peyré

April 09, 2025
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  1. Entropic Regularization Numerical Optimal Transport Gabriel Peyré É C O

    L E N O R M A L E S U P É R I E U R E www.numerical-tours.com http://optimaltransport.github.io
  2. Overview • Entropic Regularization and Sinkhorn • Convergence Analysis •

    Sinkhorn Divergences • Generative Model Fitting
  3. Schrodinger Lazy Gaz Problem 7.6. Dynamic Formulation over the Paths

    Space 111 Á = 0 Á = .05 Á = 0.2 Á = 1 Figure 7.5: Samples from Brownian bridge paths associated to the Schrödinger entropic interpola- tion (7.20) over path space. Blue corresponds to t = 0 and red to t = 1. We refer to the review paper by Léonard [2014] for an overview of this problem and an historical account of the work of Schrödinger [1931]. One can show that the (unique) ı
  4. Entropic Regularization Schr¨ odinger’s problem: [1931] Erwin Schrödinger min P1=a,P>1=b

    P i,j d(xi, yj)pPi,j + "Pi,j log(Pi,j) <latexit 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  5. Entropic Regularization Schr¨ odinger’s problem: [1931] Erwin Schrödinger min P1=a,P>1=b

    P i,j d(xi, yj)pPi,j + "Pi,j log(Pi,j) <latexit 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<latexit 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yj <latexit 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Pi,j > 10 3 " <latexit 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" = 0
  6. Entropic Regularization: General Case Relative-entropy: <latexit 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<latexit sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">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</latexit> <latexit

    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<latexit 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<latexit 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Schr¨ odinger’s problem: [1931] Erwin Schrödinger " ↵ ⇡" <latexit 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<latexit 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<latexit 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Wp ",p (↵, ) def. = min ⇡1=↵,⇡2= Z X2 dp(x, y)d⇡(x, y) + "KL(⇡|↵ ⌦ ) <latexit 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<latexit 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<latexit 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<latexit sha1_base64="9fViwbMwb16e+5fGKDRerpzqm1o=">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</latexit> KL(⇡|↵ ⌦ ) def. = Z X2 log ✓ d⇡ d↵d (x, y) ◆ d⇡(x, y) <latexit 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<latexit 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<latexit sha1_base64="pOUlfLhd2wuiDjFa6QDYJOKsVzE=">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</latexit> ⇡ = P i,j Pi,j xi,yj <latexit 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  7. Probabilistic Interpretation Christian Léonard Erwin Schrödinger min (X,Y ) {E(c(X,

    Y )) + "I(X, Y ) ; X ⇠ ↵, Y ⇠ } <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Mutual information <latexit 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<latexit 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<latexit 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Relative-entropy: <latexit 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<latexit 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<latexit 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Schr¨ odinger’s problem: [1931] Wp ",p (↵, ) def. = min ⇡1=↵,⇡2= Z X2 dp(x, y)d⇡(x, y) + "KL(⇡|↵ ⌦ ) <latexit 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<latexit 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<latexit 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<latexit sha1_base64="9fViwbMwb16e+5fGKDRerpzqm1o=">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</latexit> KL(⇡|↵ ⌦ ) def. = Z X2 log ✓ d⇡ d↵d (x, y) ◆ d⇡(x, y) <latexit 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<latexit 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  8. 7.6. Dynamic Formulation over the Paths Space 111 Á =

    0 Á = .05 Á = 0.2 Á = 1 Probabilistic Interpretation Christian Léonard Erwin Schrödinger min (X,Y ) {E(c(X, Y )) + "I(X, Y ) ; X ⇠ ↵, Y ⇠ } <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Mutual information <latexit 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<latexit 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<latexit 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Relative-entropy: <latexit 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<latexit 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<latexit 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Schr¨ odinger’s problem: [1931] Wp ",p (↵, ) def. = min ⇡1=↵,⇡2= Z X2 dp(x, y)d⇡(x, y) + "KL(⇡|↵ ⌦ ) <latexit 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<latexit sha1_base64="9fViwbMwb16e+5fGKDRerpzqm1o=">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</latexit>
  9. The Curse of Dimensionality <latexit sha1_base64="AAQHEaTZCo4XtKgRATbvN+r5c2g=">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</latexit> Theorem: <latexit sha1_base64="bQXa5RnTf3JiNE3Mcsaob7eZqxY=">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</latexit> requires

    n ⇠ (1/ )dimension <latexit 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[Dudley 1968] <latexit sha1_base64="vTWjvPBuVpUk9Xxz732SDCXW3lw=">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</latexit> ↵ <latexit sha1_base64="vTWjvPBuVpUk9Xxz732SDCXW3lw=">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</latexit> ↵ <latexit sha1_base64="kAppDnYCtZIqp/kHZtYnyPO+LpM=">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</latexit> n = 1 n P i xi <latexit sha1_base64="8rJS+aIgevIFAyo1RoikENYqprs=">AAAC3HicjVHLSsNAFD2Nr1pfURcu3ARbwVVJimiXRTcuK9gHtLUk6bQG8yKZCCV0507c+gNu9XvEP9C/8M6YglpEJyQ5c+49Z+6da4WuE3Ndf80pc/MLi0v55cLK6tr6hrq51YyDJLJZww7cIGpbZsxcx2cN7nCXtcOImZ7lspZ1fSrirRsWxU7gX/BxyHqeOfKdoWObnKi+ulPqSpPUchM26VqMm/20Oin11aJe1uXSZoGRgSKyVQ/UF3QxQAAbCTww+OCEXZiI6enAgI6QuB5S4iJCjowzTFAgbUJZjDJMYq/pO6JdJ2N92gvPWKptOsWlNyKlhn3SBJQXERanaTKeSGfB/uadSk9R25j+VublEctxRexfumnmf3WiF44hqrIHh3oKJSO6szOXRN6KqFz70hUnh5A4gQcUjwjbUjm9Z01qYtm7uFtTxt9kpmDF3s5yE7yLKmnAxs9xzoJmpWwclQ/PK8XaSTbqPHaxhwOa5zFqOEMdDVn/I57wrFwqt8qdcv+ZquQyzTa+LeXhA7QtmQM=</latexit> 8 <latexit sha1_base64="qIMVGIw1nrs/iQEzu5dZKAhRR4M=">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</latexit> 23 <latexit sha1_base64="PHdHdo8XGeIr6zZ1ehfD3lTUFPQ=">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</latexit> E|Wp(↵, n) Wp(↵, 1)| 6 Richard Dudley
  10. The Curse of Dimensionality <latexit sha1_base64="AAQHEaTZCo4XtKgRATbvN+r5c2g=">AABEjXictVxbdxTHEW6cmyEXQ/KYl3FkcnCOTATGcRwnOQatEDICBLsS2Cxw9jJaLczuLHuTYK2fktfkPX8gvyP/IHnKX0hduqd7dnumehSiOdL29PZXVV3TXV1V3aP2KOlPphsb/zz33ve+/4Mf/uj98xd+/JOf/uyDi5d+fjBJZ+NOvN9Jk3T8pN2axEl/GO9P+9MkfjIax61BO4kft19t4veP5/F40k+HjembUfxs0OoN+4f9TmsKVS8uXmpO4xPALRpHcTqOB384fXFxbePqBv1Eq4VrurCm9M9eeunDv6um6qpUddRMDVSshmoK5US11ASup+qa2lAjqHumFlA3hlKfvo/VqboA2Bm0iqFFC2pfwd8e3D3VtUO4R5oTQneASwK/Y0BG6jJgUmg3hjJyi+j7GVHG2iLaC6KJsr2Bz7amNYDaqTqCWglnWobisC9Tdah+T33oQ59GVIO962gqM9IKSh45vZoChRHUYbkL34+h3CGk0XNEmAn1HXXbou//RS2xFu87uu1M/ZukvAxXpOq692lGoaXmRD+ipzmD71ieBDj3gEKs+4ilY9L1gHo/hPYLqL8P1ymVjE7acC2o9rQUuQmXD7kpIrfh8iG3ReQuXD7krojcg8uH3NNIxI5J5358HS4fvi5yfgiXD/lQRD6Cy4d8JCIP4PIhD0Tkt3D5kN+KyNtw+ZC3ReRduHzIuyKyAZcP2RCR+3D5kPsicgsuH3JLI4tn6hiulOj0hVl5E8p5HmgpEqi5Kcp3i6yjD3srYE53CrDyrK7Bpx9bC9BpXIDdChh3hwVYeeRtg430Y2VbdIdWEx/2jojdgRHgx+6I2K/VywLs1wEz7VUBVp5ru9DOj5Wt7z2482Pvidj7UPJj5TXqAdT4sQ8CVoxRAXZPxD5UrwuwIVZ/XICV7X4d7IofK69TDWjvx4ZY01kBVranB+DB+LHyavUYav3YxyL2iTopwD4Rsd+AdfdjvwlYYd8WYM0ae4FWkB75IzHM2DJqrWxWYmkE1FoC/yRbWxLyjdtQL2F6GaZHmIGI2M4Q24GI3QyxGyzXJLOjE/J3ZS71DFEPRLSztQlLU7F9N2uPpSQAUcsQtSVEmUeKz9r0ZU7ehamRkNNs5cJSSJ/SzH5jKdbjodzyGsSDHILH9hGN/HWKljCCQk2VUTvK1nhGRnRfhjim6M300vCQcdPMKrioExHV9qDaIuqNB/VGRM08qJmImntQcxFlZ76LawaMAKt/fBYLuuMRwD5y8RWBV3ATVp07MEcjGD974AU+opoH8Fmn2Fu6yiTDaB7XScxyPMtZ4jGUFmoN6m1UWKP4OqEZFoNk3PKBjvHxDnMbCz3n2AqfZit5lGVMwun0SZ5eRge9xYjmUzU6d6nmlLw7LlXD38nmvSlVw2+Rxk/Ji+dSNfxUSz89g+wNjW2cAVuH2TTS2rflqjQ4/8I0TPkCrbpocfGpDvSYQXonFenv6Cezc4bnskkl1o8tV6Mxcfo3yfWvCg2r54mj52pU0Htir9eUoso9Geq415arypDSKjrUcti7qk8G23T1kzHlajT2wOPapJh74ZSrjt5R1htbrkbjQHHe85Q8eVOuRqNH96wPW65GA7MtLR3n23JVy44a4NjZlqta9SFlgTEHxGOea6xXNCY/aaap9ck/KM/WuD7/6jqGOZvnWYxQTsn6tsV02tlaVi6R8RdisGrTinKgfzFzfLA8jYW6LsZXLMM0t76v0rFrPGp+F7QYweznPQApZ56AhCYngdY7AYrXxKgr3zODuy7icJQcLqGaunYqeouWL2eN8nUvqFaKy2xvrR6bZK8nNPZG5BPukmYlPewWPuEiipKGdnMakulV0d1bPV/z2t8QcaMlxCgbaR3aEeKdtPI41af1uqPjy3qXZwoX7/nY8YvZ5kNtbTDmSckWoSxlPN12Jo/k1uG6uq5sjpu/i+iJor2ak9Xo047URIxCTbaYvfEF3Vva+7QnhzyYRgeeY6SpjBTvmmEWHfPpEVlU195KvFFfJkPH5QlZXWOPy9E9B93zoKvHOJuwYtyHUgNihn24awREORcyXaWk8bH6JNsdTekJlkf0Sc5CGhpsb+KchSyLso9yVI4BjaOBo/RwGst0DL65QkmO+n3y2Ng1b/kv086t2d9u0RgvHs3FmZgucb1OXCOaNbyry3fLHFiChfeb6+S/lvcS+VXhiDZU4vrc4cx6GdKOf0wR7Ig844RmmzQ78q3d/NTyN4bTnjJ757ibnZKFjMj+RbA+pTQmI/p1zw6YHXS2CAnZyBC708+8G5+v0xfHmPXj+opPNdjxFpMtmxF/Q9edXRMaixwx8DpwujS2jU52yReMietYW3c7t8tXH0TacxLuKGGKdqxcIf4f01/za8bJ2sqIQA3jE5hoW+d7HinFLKijFq3y5TbItHWl/CiT4bmW2q5/VqaPcpLVKOJCeXC17gLnDt0zLxwlY5J7stKG19GybC5SHi3pEXt7SFE82/2eXoFR7nVaJddozjVplPRgFEyzKMK0lbLIy3zLeeWph9Ge/F+oW13ntYYUI2UzuKwhKb8fU7TmSpnAqObx+4pmk1/r46VW5XyGNBYHzlz+Dmo/hL9GbnMfRqedswq3aAwwBXtnNcI10UqLMF63crzMyDS07L3lZ8ekaeXWnCW+ZutmY+x5ZSp7NGpOdNbClM9C46VD42WgDhu012i1aOqNJXohxhYNvVsZyq8Kt0YFyjORsuyRGVQ/QEo3lgqj2hWpyjG+Qb0VaW2ItFowW93dAHfOhyD9c315dn+Xre6Ruk2+TYc8MI5fujRL++RzmdrySI0pIOcb2r66s79JNci9TRYUKfM5TpwxvOvUoes0k/TXemVLyc5bi2DOLR3rNsbGNqn86QpyQHNiQvPSIG5Qi1jL78oRLVmkq47PEVHmv0U+Ffsd5TGz29o+kyjnT9h4k2eV5cWRwpD0L2Xedlai1x0nfo0oJpxp77oNtKo/YaTAGJNJ8HuWE3pCuMrxTgJ7tG2yn6t2infxho5EV0nqhfpTgI3hqNeOdXdsmR6bvv0GWqLW7VP3tZD5JcEcJX5n2dFr0ao20D7qYun+bLRaepXL35fpYbbE1+pjRm3cyMJGeXlMU30ZzIUlqsaFMSFcqvWiivzVJK8iM+9OhVI2rQ3lfKaBbcwRxUvSOVBE+Ly7K15v7mOhH+0Vem3CutS4RqKE2bhU5wdcS4tZqfMr6xDXni9djRJnJSpaKQx1d7Ww9pstZEzWL1FSzoZbu7I3c1GKnIVhCh3FJ3qL4kOX5pdw4d9I+aJDwzEkd1gH//am2lRb7+A0xGtd5oxmRDVoC7pLsXdL9zPfolxHrx3qLv0QDuE8+qBrSfo+raRVZWfKsuQu9XD6x2QFxioWpbctq/fB5SL3ZJVTlf70ybLJvekr8y5O1b4YDiE9yXMJ58P7GlIvDpV5p6laHwx1uQd5DlV4mHMMYc/ctq7Oy+VUrq9VLqE8eBUwOy4Ghzt/xbGKbRdiocbOE3n3HNA6HJZQN6vF/9oPw8dyqs4rlNuE3jV7GfDUuV2sM7LoD1efM5ZbyGgu5hjOM816Z70lPz/2+6JKTyp1evPu6aM/aseA4bVQnAeVpWO8O4qsvKFUcF/AJ0Oq/qP+cU5+G+F1RqNIjiqUzD5FMTXTQqZm3rj09c58FyKTpVMkU56ajSPqdCJ2U+2o2/C7mXmAVU+H8ruU/IlY//uzXag9JOthsuicOWhSXUzZD7uL1qV7e362SGI8y8tnextQg3vhu1SL53zvU3s869vI9a34DRKe6/dUqrq5iGR5d8/Oqzb0IL/zxjkg855vRGfpOYvFJ88GAXuLfH6KIyTz1vOCEF2KB5clXRDCjJYyym0v5TadRYoLaLdzfevQCB/pHX7cb8Bz+a0sqxSp31JdS68OuFJLUu15pHpKGYE26X8DIrTP1Dp8ruuyX9K9FUkn9AzyEp0435WfADv1jgv7FuNlyn+ZDN1ct0spmre7huUZ2FohFz7pXo7vleB7jpR1elqvKN4eq/Kc4ayE5kzL5O7jDpXJd7IeMJptZeOjPH6el/CaB/T/biH6riPpNsjSpix7RPt4Y6KXaN1skfR8nrI8X3unRFrztibTtCcq7TgwZyPL9wISPe6KZz+ff5RyNHEBHXeu80lMaXdClkeWJkQWicpMlGQW8I7wPECWeQCdQ0GaQ5FCT5REz2D6vxxf0E/Ehc9v6MIX17L/y3Fw/eq131399OGNta9u6f/Q8b76pfqVugKr0+fqKxihe2ofOB2rv6i/qr/VPqh9Vvtj7c/c9L1zGvMLlfupbf8Xi60zhA==</latexit> Theorem: <latexit sha1_base64="bQXa5RnTf3JiNE3Mcsaob7eZqxY=">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</latexit> requires

    n ⇠ (1/ )dimension <latexit 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[Dudley 1968] <latexit sha1_base64="vTWjvPBuVpUk9Xxz732SDCXW3lw=">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</latexit> ↵ <latexit sha1_base64="vTWjvPBuVpUk9Xxz732SDCXW3lw=">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</latexit> ↵ <latexit sha1_base64="kAppDnYCtZIqp/kHZtYnyPO+LpM=">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</latexit> n = 1 n P i xi <latexit sha1_base64="8rJS+aIgevIFAyo1RoikENYqprs=">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</latexit> 8 <latexit sha1_base64="qIMVGIw1nrs/iQEzu5dZKAhRR4M=">AAAC3XicjVHLSsNAFD2Nr1pfUTeCm2AruCppFXVZdOOygn1AW0KSTmswL5KJUELduRO3/oBb/R3xD/QvvDOmoBbRCUnOnHvPmbn3WqHrxFzXX3PKzOzc/EJ+sbC0vLK6pq5vNOMgiWzWsAM3iNqWGTPX8VmDO9xl7TBipme5rGVdnYp465pFsRP4F3wUsp5nDn1n4NgmJ8pQt0pdaZJabsLGXYtx00ir++OSoRb1si6XNg0qGSgiW/VAfUEXfQSwkcADgw9O2IWJmJ4OKtAREtdDSlxEyJFxhjEKpE0oi1GGSewVfYe062SsT3vhGUu1Tae49Eak1LBLmoDyIsLiNE3GE+ks2N+8U+kp7jaiv5V5ecRyXBL7l26S+V+dqIVjgGNZg0M1hZIR1dmZSyK7Im6ufamKk0NInMB9ikeEbamc9FmTmljWLnpryvibzBSs2NtZboJ3cUsacOXnOKdBs1quHJYPzqvF2kk26jy2sYM9mucRajhDHQ3yvsEjnvCsGMqtcqfcf6YquUyziW9LefgATcSZOg==</latexit> 23 <latexit sha1_base64="PHdHdo8XGeIr6zZ1ehfD3lTUFPQ=">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</latexit> E|Wp(↵, n) Wp(↵, 1)| 6 Aude Genevay <latexit sha1_base64="AAQHEaTZCo4XtKgRATbvN+r5c2g=">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</latexit> Theorem: <latexit sha1_base64="bqML1+xf+kMBSZuJMdXq4/xbgEk=">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</latexit> [Genevay 2019] <latexit 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E|W" p (↵, n) W" p (↵, 1)| 6 <latexit sha1_base64="bK09BEGbce3aga+hK85roGLmMAY=">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</latexit> requires n ⇠ (1/")dimension ⇥ (1/ )2 Richard Dudley
  11. Impact of Regularization ↵ ↵ ⇡" ⇡" ⇡" "!0 !

    ⇡ OT ⇡" "!+1 ! ↵ ⌦ Theorem: " 10 0.1 10 3
  12. Impact of Regularization ↵ ↵ ⇡" ⇡" ⇡" "!0 !

    ⇡ OT ⇡" "!+1 ! ↵ ⌦ Theorem: " 10 0.1 10 3
  13. " = 10 " = .1 " = 10 3

    ↵ ⇡" Impact of Regularization Theorem: C⇡(x, y) def. = R x 1 R y 1 d⇡(x, y) Copula: Cumulative: ⇡" (s, t) (independence) (dependence) ⇡" s t 0 1 1 " 10 0.1 10 3 " ! 0 " ! +1 st min(s, t) ⇡(s, t) def. = C⇡(C 1 ↵ (s), C 1(t))
  14. " = 10 " = .1 " = 10 3

    ↵ ⇡" Impact of Regularization Theorem: C⇡(x, y) def. = R x 1 R y 1 d⇡(x, y) Copula: Cumulative: ⇡" (s, t) (independence) (dependence) ⇡" s t 0 1 1 " 10 0.1 10 3 " ! 0 " ! +1 st min(s, t) ⇡(s, t) def. = C⇡(C 1 ↵ (s), C 1(t))
  15. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">AABB1nictVzbchu5EYU3l107N29SecrLJFqnvFuOV1a2KpetVK0sybLWtE2blC+7tF28jGjaQw7NIWXZXFWeUnnNL+Q1+Yh8R/4gecovpC/AAENipjGKY5QkDIjT3egBGt0N0L1pMsrmm5v/PPfet779ne++/8H5C9/7/g9++KOLH/74QZYuZv34sJ8m6exRr5vFyWgSH85H8yR+NJ3F3XEviR/2Xu7g5w+P41k2Sift+Ztp/GTcHU5GR6N+dw5Nzy7+tDOPTwC3bM7SaZqNsPX3p88ubmxe3aR/0Xrlmq5sKP2vmX4YHaqOGqhU9dVCjVWsJmoO9UR1VQbla3VNbaoptD1RS2ibQW1En8fqVF0A7AJ6xdCjC60v4fcQnr7WrRN4RpoZofvAJYGfGSAjdQkwKfSbQR25RfT5gihjaxntJdFE2d7A356mNYbWuXoOrRLO9AzF4Vjm6kj9lsYwgjFNqQVH19dUFqQVlDxyRjUHClNow/oAPp9BvU9Io+eIMBmNHXXbpc//RT2xFZ/7uu9C/ZukvAQlUi09+jSn0FXHRD+it7mAz1ieBDgPgUKsx4i116TrMY1+Av2X0H4HyinVjE56UJbUelqJ3IHiQ+6IyH0oPuS+iGxA8SEbIrIJxYdsaiRiZ6RzP74FxYdviZzvQfEh74nI+1B8yPsi8gEUH/KBiPwKig/5lYi8AcWHvCEib0HxIW+JyDYUH7ItIg+h+JCHInIPig+5p5HlK3UGJSU6I2FVbkO9yAMtRQIt26J818k6+rDXA9Z0vwQrr+pd+OvH7gboNC7B7gXMu6MSrDzz9sFG+rGyLbpJu4kPe1PEHsAM8GMPROyX6kUJ9suAlfayBCuvtQb082Nl63sbnvzY2yL2DtT8WHmPugstfuzdgB1jWoJtith76lUJNsTqz0qwst1vgV3xY+V9qg39/dgQa7oowcr29AF4MH6svFs9hFY/9qGIfaROSrCPROxjsO5+7OOAHfZtCdbssRdoBxmSPxLDiq2i1s1XJdamQK0r8E/yvSUh37gH7RJmmGOGhBmLiP0csR+IaOSIRrBcWW5HM/J3ZS6tHNEKRPTyvQlrc7H/IO+PtSQAsZsjdlcQVR4pvmszlmPyLkyLhJznOxfWQsaU5vYba7GeD9WW1yDuFhA8t5/TzL9C0RJGUKipKmrP8z2ekRE9VyFeU/RmRml4yLh5bhVc1ImI6nlQPRH1xoN6I6IWHtRCRB17UMciyq58F9cJmAFW//gulvTEM4B95PISgVewDbvOTVijEcyfJniB96nlLvxtUewtlSrJMJrHfRKzHE8KlngGtaXagHYbFe5SfJ3QCotBMu55V8f4+IS5jaVec2yFT/OdPMozJuF0RiTPMKeD3mJE66kenVvUckreHdfq4W/m697U6uH3SOOn5MVzrR5+rqWfn0H2tsa2z4BtwWqaau3bel0anH9hGqZ+gXZdtLj4Vsd6ziC9k5r0D/SbOTjDe9mhGuvH1uvRyJzxZYXx1aFh9Zw5eq5HBb0n9npNLao9komOe229rgwp7aITLYd9qvtmsM9AvxlTr0ejCR7XDsXcS6ded/ZO89HYej0aDxTnPU/Jkzf1ejSG9Mz6sPV6NDDb0tVxvq3XteyoAY6dbb2uVZ9QFhhzQDznucV6RTPykxaa2oj8g+psjevzr+9jmLN5mscI1ZSsb1tOp5fvZdUSGX8hBqs2rykH+hcLxwcr0liqLTG+Yhnmhf19nY7d41HzDdBiBKufzwCknHkCEpqcBFrvBCheE6Ou4sgMbkvE4Sw5WkF1dOtc9BYtX84aFdueUasUl9nRWj12yF5nNPem5BM2SLOSHhqlb7iMoqShRkFDMr06unur12tR+5sibrqCmOYzrU8nQnySVh2n+rTecnR8SZ/yzKHwmY+dv5htPtLWBmOelGwRylLF0+1n8khuG+6rV5TNcfNnEb1RtFfHZDVGdCKViVGoyRazN76kZ0v7kM7kkAfT6MN7jDSVqeJTM8yiYz49Iovq2luJN+rLZOi4npHVNfa4Gj100EMPun6MswM7xh2otSFmOISndkCUcyHXVUoan6lf5aejKb3B6og+KVhIQ4PtTVywkFVR9vMCldeAxtnAUXo4jVU6Bt9ZoyRH/T55bOxatPyX6OTWnG93aY6Xz+byTMyAuG4R14hWDZ/q8tMqB5Zg6f1ki/zX6lEivzoc0YZKXJ86nFkvEzrxjymCnZJnnNBqk1ZHsbebn1r9xHBqKnN2jqfZKVnIiOxfBPtTSnMyoh/37oA5QWeLkJCNDLE7o9y78fk6I3GOWT9upPhWg51vMdmyBfE3dN3VldFc5IiB94HTlbltdNIgXzAmrjNt3e3art59EGnvSbizhCnauXKZ+H9Mv82PmScbazMCNYxvINO2zvc+UopZUEdd2uWrbZDp60r5US7DUy213f+sTB8VJNuliAvlwd16AJz79My8cJbMSO5srQ/vo1XZXKQ8XdEjjvaIoni2+0O9A6PcV2iX3KA116FZMoRZMM+jCNNXyiKv8q3mVaQeRjv7v1C3ui5qDSlGymZwWUNSfj+maM2VMoFZzfP3Ja0mv9ZnK72q+UxoLo6dtfwNtP4cfhu5zXMYnV7BKlynOcAU7JPVCLdEaz3CeF0v8DIz09Cyz5afnZOml9tylviarZuNsY9rU2nSrDnRWQtTPwuNFw6NF4E6bNNZo9WiaTeW6JkYW7T1aWUovzrc2jUoL0TKskdmUKMAKd1YKozqQKQqx/gG9VaktSnS6sJqdU8D3DUfgvSv9dXV/U2+u0fqBvk2ffLAOH4Z0Codkc9lWqsjNaaAnD/T9tVd/R1qQe49sqBIme9x4orhU6c+ldNc0l/qnS0lO28tgrm39Fr3MTa2Q/VfryHHtCYyWpcG8Rn1iLX8rhzRikW66vgcEWX+u+RTsd9RHTO7ve07iQr+hI03eVVZXhwpTEj/UubtYC16PXDi14hiwoX2rntAq/4bRgqMMZkEv2eZ0RvCXY5PEtij7ZH9XLdTfIo3cSS6SlIv1R8CbAxHvXauu3PLjNiM7RPoiVq3b93XQ+aXBHOU+J3lRK9Lu9pY+6jLleez0erqXa74XKWHxQpfq48F9XEjCxvlFTEd9XkwF5aoHhfGhHCpN4o68teTvI7MfDoVStn0NpSLmQa2Mc8pXpLugSLC591d9npzHwvj6K3R6xHWpcYtEiXMxqU6P+BaWsxKnV/bh7j1fOVulDg7UdlOYai7u4W132whY7J+iZJyNtzblb1TiFLkLAxT6Cu+0VsWH7o0P4eCvyPliw4Nx5DcYQv82221o/bewW2IV7rOGc2IWtAWDFZi764eZ7FHtY5eOdRd+iEcwnmMQNeS9CPaSevKzpRlyV3q4fRfkxWYqViU3vasPwaXizySdU51xjMiyyaPZqTMd3HqjsVwCBlJkUs4Hz7XkEZxpMx3muqNwVCXR1DkUIeHuccQ9s5t7/q8XE7V+lrnEsqDdwFz4mJwePJXHqvYfiEWaua8kXfPAa3DUQV1s1v8r+MwfCyn+rxCuWX0XbMXAW+d+8U6I4v+cP01Y7mFzOZyjuE803x01lvy82O/L6r1plJnNO+ePvqjdg4YXkvFeVBZOsa7s8jKG0oFzwV8MqTqP+of5+RvI7zKaZTJUYeSOacop2Z6yNTMNy59ozOfhchk6ZTJVKRm44gW3YjdUQfqBvzs5B5g3duh/F1K/otY//dnB9B6RNbDZNE5c9ChtpiyH/YUbUDP9v5smcR4l5fv9rahBc/CG9SK93zvUH+869sujK38GyS81m+rVA0KEcnq6Z5dVz0YQfHkjXNA5nu+Ed2l5ywW3zwbB5wtmvtTqxIt6RP5ZkGvFN9zpOzTXJ3qs3o8OcAb9t08PxSpT6mtq+087rkS52Yp5+YK54y0U+Rw4nxWfTerjMuOw2WQ586Odb+U4mx7nledG90t5cJ30Kvxwwr80JGyRdp/SZHwTFVn8xYVNBdaJveEdaJMJpL1gHFmN3/f1ZHtcQWv44Dx3ypF33Ik3QdZepT/juiEbUb0Eq2bPZKebzpWZ1JvVkirv0f57OLGtdX/y2C9crh19XdXr93b2vjiuv5vDj5QP1O/UJdhif9GfQHEmuqQDr3/qv6m/r79ePuP23/a/jN3fe+cxvxEFf5t/+W/mrqpxQ==</latexit> <latexit 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<latexit

    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<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="TzmMHppjQPGpkFJfRm72fbNVqls=">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</latexit> min P nP i,j d(xi, yj)pPi,j + "Pi,j log(Pi,j) ; P1 = a, P>1 = b o <latexit 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<latexit 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<latexit sha1_base64="tQGZ+FtykZ5fhP2XyYtnLKz3TZM=">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</latexit> Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tM9ZoNF+/Iw0I2pZoWiOJ0AgbhI=">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</latexit>
  16. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit 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<latexit

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<latexit 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u (Kv) = a <latexit 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sha1_base64="L0a9yaq7SoDrCaDqUKFsvOMilSM=">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</latexit> v (K>u) = b <latexit 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<latexit 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<latexit sha1_base64="UEfo0q7/awH1ZlqoxebmYfYkUMI=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="TzmMHppjQPGpkFJfRm72fbNVqls=">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</latexit> min P nP i,j d(xi, yj)pPi,j + "Pi,j log(Pi,j) ; P1 = a, P>1 = b o <latexit 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<latexit 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<latexit sha1_base64="tQGZ+FtykZ5fhP2XyYtnLKz3TZM=">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</latexit> Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tM9ZoNF+/Iw0I2pZoWiOJ0AgbhI=">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</latexit> P = diag(u)K diag(v) <latexit 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<latexit 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=) <latexit 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<latexit sha1_base64="0h78JjXFym7avQJT/Y7NxpEtnC0=">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</latexit> a = P1 = diag(u)(Kv) = u (Kv)
  17. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">AABB1nictVzbchu5EYU3l107N29SecrLJFqnvFuOV1a2KpetVK0sybLWtE2blC+7tF28jGjaQw7NIWXZXFWeUnnNL+Q1+Yh8R/4gecovpC/AAENipjGKY5QkDIjT3egBGt0N0L1pMsrmm5v/PPfet779ne++/8H5C9/7/g9++KOLH/74QZYuZv34sJ8m6exRr5vFyWgSH85H8yR+NJ3F3XEviR/2Xu7g5w+P41k2Sift+Ztp/GTcHU5GR6N+dw5Nzy7+tDOPTwC3bM7SaZqNsPX3p88ubmxe3aR/0Xrlmq5sKP2vmX4YHaqOGqhU9dVCjVWsJmoO9UR1VQbla3VNbaoptD1RS2ibQW1En8fqVF0A7AJ6xdCjC60v4fcQnr7WrRN4RpoZofvAJYGfGSAjdQkwKfSbQR25RfT5gihjaxntJdFE2d7A356mNYbWuXoOrRLO9AzF4Vjm6kj9lsYwgjFNqQVH19dUFqQVlDxyRjUHClNow/oAPp9BvU9Io+eIMBmNHXXbpc//RT2xFZ/7uu9C/ZukvAQlUi09+jSn0FXHRD+it7mAz1ieBDgPgUKsx4i116TrMY1+Av2X0H4HyinVjE56UJbUelqJ3IHiQ+6IyH0oPuS+iGxA8SEbIrIJxYdsaiRiZ6RzP74FxYdviZzvQfEh74nI+1B8yPsi8gEUH/KBiPwKig/5lYi8AcWHvCEib0HxIW+JyDYUH7ItIg+h+JCHInIPig+5p5HlK3UGJSU6I2FVbkO9yAMtRQIt26J818k6+rDXA9Z0vwQrr+pd+OvH7gboNC7B7gXMu6MSrDzz9sFG+rGyLbpJu4kPe1PEHsAM8GMPROyX6kUJ9suAlfayBCuvtQb082Nl63sbnvzY2yL2DtT8WHmPugstfuzdgB1jWoJtith76lUJNsTqz0qwst1vgV3xY+V9qg39/dgQa7oowcr29AF4MH6svFs9hFY/9qGIfaROSrCPROxjsO5+7OOAHfZtCdbssRdoBxmSPxLDiq2i1s1XJdamQK0r8E/yvSUh37gH7RJmmGOGhBmLiP0csR+IaOSIRrBcWW5HM/J3ZS6tHNEKRPTyvQlrc7H/IO+PtSQAsZsjdlcQVR4pvmszlmPyLkyLhJznOxfWQsaU5vYba7GeD9WW1yDuFhA8t5/TzL9C0RJGUKipKmrP8z2ekRE9VyFeU/RmRml4yLh5bhVc1ImI6nlQPRH1xoN6I6IWHtRCRB17UMciyq58F9cJmAFW//gulvTEM4B95PISgVewDbvOTVijEcyfJniB96nlLvxtUewtlSrJMJrHfRKzHE8KlngGtaXagHYbFe5SfJ3QCotBMu55V8f4+IS5jaVec2yFT/OdPMozJuF0RiTPMKeD3mJE66kenVvUckreHdfq4W/m697U6uH3SOOn5MVzrR5+rqWfn0H2tsa2z4BtwWqaau3bel0anH9hGqZ+gXZdtLj4Vsd6ziC9k5r0D/SbOTjDe9mhGuvH1uvRyJzxZYXx1aFh9Zw5eq5HBb0n9npNLao9komOe229rgwp7aITLYd9qvtmsM9AvxlTr0ejCR7XDsXcS6ded/ZO89HYej0aDxTnPU/Jkzf1ejSG9Mz6sPV6NDDb0tVxvq3XteyoAY6dbb2uVZ9QFhhzQDznucV6RTPykxaa2oj8g+psjevzr+9jmLN5mscI1ZSsb1tOp5fvZdUSGX8hBqs2rykH+hcLxwcr0liqLTG+Yhnmhf19nY7d41HzDdBiBKufzwCknHkCEpqcBFrvBCheE6Ou4sgMbkvE4Sw5WkF1dOtc9BYtX84aFdueUasUl9nRWj12yF5nNPem5BM2SLOSHhqlb7iMoqShRkFDMr06unur12tR+5sibrqCmOYzrU8nQnySVh2n+rTecnR8SZ/yzKHwmY+dv5htPtLWBmOelGwRylLF0+1n8khuG+6rV5TNcfNnEb1RtFfHZDVGdCKViVGoyRazN76kZ0v7kM7kkAfT6MN7jDSVqeJTM8yiYz49Iovq2luJN+rLZOi4npHVNfa4Gj100EMPun6MswM7xh2otSFmOISndkCUcyHXVUoan6lf5aejKb3B6og+KVhIQ4PtTVywkFVR9vMCldeAxtnAUXo4jVU6Bt9ZoyRH/T55bOxatPyX6OTWnG93aY6Xz+byTMyAuG4R14hWDZ/q8tMqB5Zg6f1ki/zX6lEivzoc0YZKXJ86nFkvEzrxjymCnZJnnNBqk1ZHsbebn1r9xHBqKnN2jqfZKVnIiOxfBPtTSnMyoh/37oA5QWeLkJCNDLE7o9y78fk6I3GOWT9upPhWg51vMdmyBfE3dN3VldFc5IiB94HTlbltdNIgXzAmrjNt3e3art59EGnvSbizhCnauXKZ+H9Mv82PmScbazMCNYxvINO2zvc+UopZUEdd2uWrbZDp60r5US7DUy213f+sTB8VJNuliAvlwd16AJz79My8cJbMSO5srQ/vo1XZXKQ8XdEjjvaIoni2+0O9A6PcV2iX3KA116FZMoRZMM+jCNNXyiKv8q3mVaQeRjv7v1C3ui5qDSlGymZwWUNSfj+maM2VMoFZzfP3Ja0mv9ZnK72q+UxoLo6dtfwNtP4cfhu5zXMYnV7BKlynOcAU7JPVCLdEaz3CeF0v8DIz09Cyz5afnZOml9tylviarZuNsY9rU2nSrDnRWQtTPwuNFw6NF4E6bNNZo9WiaTeW6JkYW7T1aWUovzrc2jUoL0TKskdmUKMAKd1YKozqQKQqx/gG9VaktSnS6sJqdU8D3DUfgvSv9dXV/U2+u0fqBvk2ffLAOH4Z0Codkc9lWqsjNaaAnD/T9tVd/R1qQe49sqBIme9x4orhU6c+ldNc0l/qnS0lO28tgrm39Fr3MTa2Q/VfryHHtCYyWpcG8Rn1iLX8rhzRikW66vgcEWX+u+RTsd9RHTO7ve07iQr+hI03eVVZXhwpTEj/UubtYC16PXDi14hiwoX2rntAq/4bRgqMMZkEv2eZ0RvCXY5PEtij7ZH9XLdTfIo3cSS6SlIv1R8CbAxHvXauu3PLjNiM7RPoiVq3b93XQ+aXBHOU+J3lRK9Lu9pY+6jLleez0erqXa74XKWHxQpfq48F9XEjCxvlFTEd9XkwF5aoHhfGhHCpN4o68teTvI7MfDoVStn0NpSLmQa2Mc8pXpLugSLC591d9npzHwvj6K3R6xHWpcYtEiXMxqU6P+BaWsxKnV/bh7j1fOVulDg7UdlOYai7u4W132whY7J+iZJyNtzblb1TiFLkLAxT6Cu+0VsWH7o0P4eCvyPliw4Nx5DcYQv82221o/bewW2IV7rOGc2IWtAWDFZi764eZ7FHtY5eOdRd+iEcwnmMQNeS9CPaSevKzpRlyV3q4fRfkxWYqViU3vasPwaXizySdU51xjMiyyaPZqTMd3HqjsVwCBlJkUs4Hz7XkEZxpMx3muqNwVCXR1DkUIeHuccQ9s5t7/q8XE7V+lrnEsqDdwFz4mJwePJXHqvYfiEWaua8kXfPAa3DUQV1s1v8r+MwfCyn+rxCuWX0XbMXAW+d+8U6I4v+cP01Y7mFzOZyjuE803x01lvy82O/L6r1plJnNO+ePvqjdg4YXkvFeVBZOsa7s8jKG0oFzwV8MqTqP+of5+RvI7zKaZTJUYeSOacop2Z6yNTMNy59ozOfhchk6ZTJVKRm44gW3YjdUQfqBvzs5B5g3duh/F1K/otY//dnB9B6RNbDZNE5c9ChtpiyH/YUbUDP9v5smcR4l5fv9rahBc/CG9SK93zvUH+869sujK38GyS81m+rVA0KEcnq6Z5dVz0YQfHkjXNA5nu+Ed2l5ywW3zwbB5wtmvtTqxIt6RP5ZkGvFN9zpOzTXJ3qs3o8OcAb9t08PxSpT6mtq+087rkS52Yp5+YK54y0U+Rw4nxWfTerjMuOw2WQ586Odb+U4mx7nledG90t5cJ30Kvxwwr80JGyRdp/SZHwTFVn8xYVNBdaJveEdaJMJpL1gHFmN3/f1ZHtcQWv44Dx3ypF33Ik3QdZepT/juiEbUb0Eq2bPZKebzpWZ1JvVkirv0f57OLGtdX/y2C9crh19XdXr93b2vjiuv5vDj5QP1O/UJdhif9GfQHEmuqQDr3/qv6m/r79ePuP23/a/jN3fe+cxvxEFf5t/+W/mrqpxQ==</latexit> <latexit 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<latexit

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<latexit 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u (Kv) = a <latexit 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sha1_base64="L0a9yaq7SoDrCaDqUKFsvOMilSM=">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</latexit> v (K>u) = b <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="c65tFjuRWb0VgfFjoHmUj+uQL6Q=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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</latexit> a = P1 = diag(u)(Kv) = u (Kv)
  18. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit 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<latexit

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<latexit 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u (Kv) = a <latexit 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sha1_base64="L0a9yaq7SoDrCaDqUKFsvOMilSM=">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</latexit> v (K>u) = b <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="c65tFjuRWb0VgfFjoHmUj+uQL6Q=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="TzmMHppjQPGpkFJfRm72fbNVqls=">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</latexit> min P nP i,j d(xi, yj)pPi,j + "Pi,j log(Pi,j) ; P1 = a, P>1 = b o <latexit 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<latexit 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! Convolution on regular grids, separable kernels. <latexit 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<latexit 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<latexit 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! Parallelizable on GPUs. <latexit 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<latexit 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<latexit 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<latexit 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Matrix/vector multiplications: ! O(n2 /"2) complexity. <latexit 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<latexit 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<latexit 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<latexit 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Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tM9ZoNF+/Iw0I2pZoWiOJ0AgbhI=">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</latexit> Theorem: [Sinkhorn 1964] <latexit 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<latexit 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<latexit sha1_base64="gcU/kHyZLcweAP0wHsiW13HNtBs=">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</latexit> (u, v) converges. <latexit 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<latexit 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<latexit 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Richard Sinkhorn P = diag(u)K diag(v) <latexit 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<latexit 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=) <latexit 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<latexit sha1_base64="0h78JjXFym7avQJT/Y7NxpEtnC0=">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</latexit> a = P1 = diag(u)(Kv) = u (Kv)
  19. RAS algorithm Iterative proportional fitting Sinkhorn algorithm Biproportional fitting Matrix

    scaling DAD scaling Sinkhorn, IPFP, RAS, … Kruithof, 1937 Udny 1912 Sinkhorn 1964 Deming and Stephan, 1940 Yule Udny Edwards Deming Frederick Stephan Many names: Richard Sinkhorn Gravity model Special thanks to Tony Silveti-Falls for the picture of Sinkhorn
  20. Other Regularizations ↵ R(⇡) = Z log ✓ d⇡ dxdy

    ◆ d⇡(x, y) R(⇡) = Z ✓ d⇡ dxdy ◆2 dxdy min ⇡ ⇢Z R2 ||x y||2d⇡(x, y) + "R(⇡) ; ⇡1 = ↵, ⇡2 = constraints P1 m = a and P€1 n = b (one can a ord to ignore the no straint using entropy because that penalty incorporates a logarithmic t the entries of P to stay in the positive orthant). This implies that the s is no longer a ne and iterative Bregman projections do not converge solution. A workaround is to use instead Dykstra’s algorithm (1983 [Bauschke and Lewis, 2000]), as detailed in [Benamou et al., 2015]. uses projections according to the Bregman divergence associated to Remark 8.1 for more details regarding Bregman divergences. An issu eral these projections cannot be computed explicitly. For the squared n corresponds to computing the Euclidean projection on (C 1 a , C 2 b ) (with tivity constraints), which can be solved e ciently using projection alg plices [Condat, 2015]. The main advantage of the quadratic regularizat is that it produces sparse approximation of the optimal coupling, yet t expense of a slower algorithm that cannot be parallelized as e cientl compute several optimal transports simultaneously (as discussed in § contrasts the approximation achieved by entropic and quadratic regul
  21. Other Regularizations ↵ R(⇡) = Z log ✓ d⇡ dxdy

    ◆ d⇡(x, y) R(⇡) = Z ✓ d⇡ dxdy ◆2 dxdy min ⇡ ⇢Z R2 ||x y||2d⇡(x, y) + "R(⇡) ; ⇡1 = ↵, ⇡2 = constraints P1 m = a and P€1 n = b (one can a ord to ignore the no straint using entropy because that penalty incorporates a logarithmic t the entries of P to stay in the positive orthant). This implies that the s is no longer a ne and iterative Bregman projections do not converge solution. A workaround is to use instead Dykstra’s algorithm (1983 [Bauschke and Lewis, 2000]), as detailed in [Benamou et al., 2015]. uses projections according to the Bregman divergence associated to Remark 8.1 for more details regarding Bregman divergences. An issu eral these projections cannot be computed explicitly. For the squared n corresponds to computing the Euclidean projection on (C 1 a , C 2 b ) (with tivity constraints), which can be solved e ciently using projection alg plices [Condat, 2015]. The main advantage of the quadratic regularizat is that it produces sparse approximation of the optimal coupling, yet t expense of a slower algorithm that cannot be parallelized as e cientl compute several optimal transports simultaneously (as discussed in § contrasts the approximation achieved by entropic and quadratic regul
  22. Overview • Entropic Regularization and Sinkhorn • Convergence Analysis •

    Sinkhorn Divergences • Generative Model Fitting
  23. Bregman Iterative Projections the kernel K become too negligible to

    be stored in memory as posit become instead null. This can then result in a matrix product Kv smaller entries that become null and result in a division by 0 in the of Equation (4.15). Such issues can be partly resolved by carrying on the multipliers u and v in the log domain. That approach is care Remark 4.22, and related to a direct resolution of the dual of proble Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ + Since the sets C 1 a and C 2 b are a ne, these iterations are known to conve of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhor since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has the kernel K become too negligible to be stored in memory as positive nu become instead null. This can then result in a matrix product Kv or K T u smaller entries that become null and result in a division by 0 in the Sinkh of Equation (4.15). Such issues can be partly resolved by carrying out co on the multipliers u and v in the log domain. That approach is carefully p Remark 4.22, and related to a direct resolution of the dual of problem (4.2 Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One can us iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). Since the sets C 1 a and C 2 b are a ne, these iterations are known to converge to t of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhorn itera since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has P (2 ¸ +1) def. = diag(u ( ¸ +1) )K diag(v ( ¸ ) ) hP, Ci + " KL(P|a ⌦ b) = " KL(P|K) + cst <latexit 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<latexit sha1_base64="BKDev5VQxrfCyjh/WSsW5fgsAQ4=">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</latexit> where Ki,j = e Ci,j " aibj <latexit 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<latexit 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<latexit 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Shr¨ odinger problem: <latexit 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<latexit 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min P2U(a,b) KL(P|K) <latexit 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Constraints : <latexit sha1_base64="an9unSN7xD1XRFITI4eRXKm8G4A=">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</latexit> C1 a [ C2 b
  24. Bregman Iterative Projections K = P(0) <latexit 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<latexit 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    <latexit 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the kernel K become too negligible to be stored in memory as posit become instead null. This can then result in a matrix product Kv smaller entries that become null and result in a division by 0 in the of Equation (4.15). Such issues can be partly resolved by carrying on the multipliers u and v in the log domain. That approach is care Remark 4.22, and related to a direct resolution of the dual of proble Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ + Since the sets C 1 a and C 2 b are a ne, these iterations are known to conve of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhor since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has the kernel K become too negligible to be stored in memory as positive nu become instead null. This can then result in a matrix product Kv or K T u smaller entries that become null and result in a division by 0 in the Sinkh of Equation (4.15). Such issues can be partly resolved by carrying out co on the multipliers u and v in the log domain. That approach is carefully p Remark 4.22, and related to a direct resolution of the dual of problem (4.2 Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One can us iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). Since the sets C 1 a and C 2 b are a ne, these iterations are known to converge to t of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhorn itera since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has P (2 ¸ +1) def. = diag(u ( ¸ +1) )K diag(v ( ¸ ) ) Theorem: <latexit 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<latexit 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<latexit 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<latexit 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For a ne (C1 a , C2 b ), <latexit 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<latexit 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<latexit sha1_base64="YSyjMkuIcdYOMO9YWj32dGrHkW0=">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</latexit> P? = argmin P2C1 a \C1 b KL(P|K) <latexit 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<latexit 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P(`) ! <latexit 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[Bregman, 1967] <latexit 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the kernel K become too negligible to be stored in memory as positive nu become instead null. This can then result in a matrix product Kv or K T smaller entries that become null and result in a division by 0 in the Sinkh of Equation (4.15). Such issues can be partly resolved by carrying out co on the multipliers u and v in the log domain. That approach is carefully p Remark 4.22, and related to a direct resolution of the dual of problem (4.2 Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One can u iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). Since the sets C 1 a and C 2 b are a ne, these iterations are known to converge to of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhorn itera since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has P (2 ¸ +1) def. = diag(u ( ¸ +1) )K diag(v ( ¸ ) ) and P (2 ¸ +2) def. = diag(u ( ¸ +1) )K diag(v ( ¸ +1) ) Iterative projections: <latexit 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<latexit sha1_base64="gtAbJeApPn1kIekGgqP+O3RizFI=">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</latexit> hP, Ci + " KL(P|a ⌦ b) = " KL(P|K) + cst <latexit 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<latexit 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<latexit sha1_base64="BKDev5VQxrfCyjh/WSsW5fgsAQ4=">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</latexit> where Ki,j = e Ci,j " aibj <latexit 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<latexit 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<latexit 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Shr¨ odinger problem: <latexit 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<latexit 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min P2U(a,b) KL(P|K) <latexit 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Constraints : <latexit sha1_base64="an9unSN7xD1XRFITI4eRXKm8G4A=">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</latexit> C1 a [ C2 b
  25. Bregman Iterative Projections K = P(0) <latexit 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<latexit 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    <latexit 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<latexit sha1_base64="xVN+FpNm8xwt23lxDkEcr+0vbvc=">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</latexit> the kernel K become too negligible to be stored in memory as posit become instead null. This can then result in a matrix product Kv smaller entries that become null and result in a division by 0 in the of Equation (4.15). Such issues can be partly resolved by carrying on the multipliers u and v in the log domain. That approach is care Remark 4.22, and related to a direct resolution of the dual of proble Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ + Since the sets C 1 a and C 2 b are a ne, these iterations are known to conve of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhor since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has the kernel K become too negligible to be stored in memory as positive nu become instead null. This can then result in a matrix product Kv or K T u smaller entries that become null and result in a division by 0 in the Sinkh of Equation (4.15). Such issues can be partly resolved by carrying out co on the multipliers u and v in the log domain. That approach is carefully p Remark 4.22, and related to a direct resolution of the dual of problem (4.2 Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One can us iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). Since the sets C 1 a and C 2 b are a ne, these iterations are known to converge to t of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhorn itera since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has P (2 ¸ +1) def. = diag(u ( ¸ +1) )K diag(v ( ¸ ) ) practice, however, that slowdown is rarely observed in practice for a simpler reason: Sinkhorn’s algorithm will often fail to terminate as soon as some of the elements of the kernel K become too negligible to be stored in memory as positive numbers, and become instead null. This can then result in a matrix product Kv or K T u with ever smaller entries that become null and result in a division by 0 in the Sinkhorn update of Equation (4.15). Such issues can be partly resolved by carrying out computations on the multipliers u and v in the log domain. That approach is carefully presented in Remark 4.22, and related to a direct resolution of the dual of problem (4.2). Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One can use Bregman iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). (4.16) Since the sets C 1 a and C 2 b are a ne, these iterations are known to converge to the solution of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhorn iterations (4.15) since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has P (2 ¸ +1) def. = diag(u ( ¸ +1) )K diag(v ( ¸ ) ) and P (2 ¸ +2) def. = diag(u ( ¸ +1) )K diag(v ( ¸ +1) ) l Stability of Sinkhorn Iterations). As we discuss in Remarks 4.13 nce of Sinkhorn’s algorithm deteriorates as Á æ 0. In numerical t slowdown is rarely observed in practice for a simpler reason: will often fail to terminate as soon as some of the elements of oo negligible to be stored in memory as positive numbers, and This can then result in a matrix product Kv or K T u with ever come null and result in a division by 0 in the Sinkhorn update ch issues can be partly resolved by carrying out computations nd v in the log domain. That approach is carefully presented in ted to a direct resolution of the dual of problem (4.2). with iterative projections). Denoting P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô constraints, one has U(a, b) = C 1 a fl C 2 b . One can use Bregman Bregman, 1967] f. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). (4.16) 2 b are a ne, these iterations are known to converge to the solution 1967]. These iterates are equivalent to Sinkhorn iterations (4.15) P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), Theorem: <latexit 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<latexit 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<latexit 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<latexit 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For a ne (C1 a , C2 b ), <latexit 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<latexit 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<latexit sha1_base64="YSyjMkuIcdYOMO9YWj32dGrHkW0=">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</latexit> P? = argmin P2C1 a \C1 b KL(P|K) <latexit 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<latexit 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P(`) ! <latexit 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[Bregman, 1967] <latexit 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the kernel K become too negligible to be stored in memory as positive nu become instead null. This can then result in a matrix product Kv or K T smaller entries that become null and result in a division by 0 in the Sinkh of Equation (4.15). Such issues can be partly resolved by carrying out co on the multipliers u and v in the log domain. That approach is carefully p Remark 4.22, and related to a direct resolution of the dual of problem (4.2 Remark 4.7 (Relation with iterative projections). Denoting C 1 a def. = {P : P1 m = a} and C 2 b def. = Ó P : P T 1 m = b Ô the rows and columns constraints, one has U(a, b) = C 1 a fl C 2 b . One can u iterative projections [Bregman, 1967] P ( ¸ +1) def. = ProjKL C 1 a (P ( ¸ ) ) and P ( ¸ +2) def. = ProjKL C 2 b (P ( ¸ +1) ). Since the sets C 1 a and C 2 b are a ne, these iterations are known to converge to of (4.7), see [Bregman, 1967]. These iterates are equivalent to Sinkhorn itera since defining P (2 ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ), one has P (2 ¸ +1) def. = diag(u ( ¸ +1) )K diag(v ( ¸ ) ) and P (2 ¸ +2) def. = diag(u ( ¸ +1) )K diag(v ( ¸ +1) ) Iterative projections: <latexit 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<latexit sha1_base64="gtAbJeApPn1kIekGgqP+O3RizFI=">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</latexit> hP, Ci + " KL(P|a ⌦ b) = " KL(P|K) + cst <latexit 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<latexit 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<latexit sha1_base64="BKDev5VQxrfCyjh/WSsW5fgsAQ4=">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</latexit> where Ki,j = e Ci,j " aibj <latexit 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<latexit 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<latexit 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Shr¨ odinger problem: <latexit 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<latexit 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min P2U(a,b) KL(P|K) <latexit 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Constraints : <latexit 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C1 a [ C2 b
  26. Dual Analysis: Alternate Maximization <latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p

    def. = <latexit 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sup (f,g)2C(X)2 Z fd↵ + Z gd + " Z X2 (1 e dp+f g " )d↵ ⌦ d <latexit 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d⇡?(x, y) = ef?(x)+g?(y) c(x,y) " d↵(x)d (y) <latexit 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Primal-dual relations: <latexit sha1_base64="b5HPF1lmAzsVMyGqVH/qAOvhbiI=">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</latexit> Dual problem:
  27. Dual Analysis: Alternate Maximization Soft c-transforms: <latexit 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<latexit 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fc,"(y) def. = min ↵ "(dp(·, y) f) <latexit 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<latexit 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gc,"(x) def. = min"(dp(x, ·) g) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. = <latexit 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sup (f,g)2C(X)2 Z fd↵ + Z gd + " Z X2 (1 e dp+f g " )d↵ ⌦ d <latexit 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d⇡?(x, y) = ef?(x)+g?(y) c(x,y) " d↵(x)d (y) <latexit 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Primal-dual relations: <latexit 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Dual problem: Soft min: <latexit 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min ↵ "(h) def. = " log Z X e h/"d↵ <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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"!0 ! min supp(↵) h
  28. Dual Analysis: Alternate Maximization Soft c-transforms: <latexit 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<latexit 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fc,"(y) def. = min ↵ "(dp(·, y) f) <latexit 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<latexit 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gc,"(x) def. = min"(dp(x, ·) g) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. = <latexit sha1_base64="1fc+j8d8XBqoBLottFct+NtS058=">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</latexit> sup (f,g)2C(X)2 Z fd↵ + Z gd + " Z X2 (1 e dp+f g " )d↵ ⌦ d Sinkhorn’s algorithm: <latexit 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<latexit 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<latexit sha1_base64="5QxnzlaOeMrx2y527LIbv6KfndU=">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</latexit> fk+1 def. = (gk)c," <latexit sha1_base64="+c+nAJ/SGm0FoRamb6DVcpO7CQI=">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</latexit> gk+1 def. = (fk+1)c," Richard Sinkhorn <latexit 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d⇡?(x, y) = ef?(x)+g?(y) c(x,y) " d↵(x)d (y) <latexit 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Primal-dual relations: <latexit sha1_base64="b5HPF1lmAzsVMyGqVH/qAOvhbiI=">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</latexit> Dual problem: <latexit sha1_base64="BQAUKxzc8Qb00w1LLPj3yumkUHs=">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</latexit> Proposition: f0 = 0 6 f1 6 f2 6 . . . <latexit 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If c is bounded, fk ! f?. <latexit 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[Robert Fortet 1938] Soft min: <latexit 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min ↵ "(h) def. = " log Z X e h/"d↵ <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="wV/4AS/OBY2C8Cq5XnMy4SxHeLU=">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</latexit> "!0 ! min supp(↵) h
  29. Stabilized Sinkhorn <latexit sha1_base64="er0D6V7aL+pbEpN9U38WFu2+H5I=">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</latexit> min P hP, Ci + "hP,

    log(P)i ; P1 = a, P>1 = b P = diag(u)K diag(v) <latexit sha1_base64="VSmNhdQ9or4pXCDzkM6UsvN5TRc=">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</latexit> <latexit 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u = a e f " <latexit 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v = b e g " <latexit 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P = ef g C " <latexit sha1_base64="25V0hq5UE1QfPxJlglCNn15x6so=">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</latexit> max f,g hf, ai + hg, bi "hef g C " , a ⌦ bi
  30. Stabilized Sinkhorn <latexit sha1_base64="er0D6V7aL+pbEpN9U38WFu2+H5I=">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</latexit> min P hP, Ci + "hP,

    log(P)i ; P1 = a, P>1 = b P = diag(u)K diag(v) <latexit sha1_base64="VSmNhdQ9or4pXCDzkM6UsvN5TRc=">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</latexit> <latexit 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u = a e f " <latexit 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v = b e g " <latexit 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P = ef g C " <latexit 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gj min" a (C·,j f) <latexit 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fi min" b (Ci,· g) u a Kv <latexit 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<latexit 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<latexit 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<latexit 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  31. Stabilized Sinkhorn <latexit sha1_base64="er0D6V7aL+pbEpN9U38WFu2+H5I=">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</latexit> min P hP, Ci + "hP,

    log(P)i ; P1 = a, P>1 = b P = diag(u)K diag(v) <latexit sha1_base64="VSmNhdQ9or4pXCDzkM6UsvN5TRc=">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</latexit> <latexit 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u = a e f " <latexit 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v = b e g " <latexit 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P = ef g C " <latexit 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gj min" a (C·,j f) <latexit 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fi min" b (Ci,· g) u a Kv <latexit 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<latexit 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<latexit 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<latexit 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Unstable for small ". <latexit 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min" a (h ) = min" a (h) <latexit sha1_base64="C9A2FrBh9lOjOdGM+SI2uRkFCtI=">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</latexit> Use = min(h) or . . .
  32. Stabilized Sinkhorn <latexit sha1_base64="er0D6V7aL+pbEpN9U38WFu2+H5I=">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</latexit> min P hP, Ci + "hP,

    log(P)i ; P1 = a, P>1 = b P = diag(u)K diag(v) <latexit sha1_base64="VSmNhdQ9or4pXCDzkM6UsvN5TRc=">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</latexit> <latexit 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u = a e f " <latexit 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v = b e g " <latexit 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P = ef g C " <latexit 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gj min" a (C·,j f) <latexit 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fi min" b (Ci,· g) u a Kv <latexit 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<latexit 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<latexit 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<latexit 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Unstable for small ". <latexit 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min" a (h ) = min" a (h) <latexit 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Use = min(h) or . . . <latexit sha1_base64="O4e8JO+tB9/QdCOVHdfvByFjJCQ=">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</latexit> Proposition: during iterations f g 6 C. <latexit 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fi min" b (Ci,· g fi) + fi <latexit 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gj min" a (C·,j f gj) + gj
  33. d H is a distance on the set of rays

    ! u . ! u 0 ! u = {ru ; r > 0} dH (u,u 0 ) Hilbert’s projective metric: Hilbert Projective Metric d H (u, u0) def. = || log(u) log(u0)||V <latexit 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<latexit 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<latexit 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||f||V def. = max(f) min(f) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  34. d H is a distance on the set of rays

    ! u . ! u 0 ! u = {ru ; r > 0} dH (u,u 0 ) Hilbert’s projective metric: This can be shows to be a distance on the projective cone R+,ú/ ≥, where u ≥ uÕ means that ÷s > 0, u = suÕ (the vector are equal up to rescaling, hence the naming “projective”). This means that dH satisfies the triangular inequality and dH (u, uÕ) = 0 if and only if u ≥ uÕ. This is a projective version of Hilbert’s original distance on bounded open convex sets Hilbert [1895]. The projec- tive cone Rn +,ú/ ≥ is a complete metric space for this distance. It was introduced independently by [Birkho , 1957] and [Samel- son et al., 1957]. They proved the following fundamental theorem, which shows that a positive matrix is a strict contraction on the cone of positive vectors. Theorem 1.1. Let K œ Rn◊m +,ú , then for (v, vÕ) œ (Rm +,ú )2 dH (Kv, KvÕ) Æ ⁄(K)dH (v, vÕ) where Y _ ] _ [ ⁄(K) def. = Ô ÷(K) ≠ 1 Ô ÷(K)+1 < 1 ÷(K) def. = max i,j,k,¸ Ki,k Kj,¸ Kj,k Ki,¸ . Remark 1.11 (Perron-Frobenius). A typical application of Theo- rem 1.1 is to provide a quantitative proof of Perron-Frobenius theorem, which, as explained in Remark 1.13, is linked to a lo- cal linearization of Sinkhorn’s iterates. A matrix K œ Rn◊n + with K€1n = 1n maps n into n. If furthermore K > 0, then ac- cording to Theorem 1.1, it is strictly contractant for the met- Birkho↵’s contraction theorem: R2 + KR2 + K 2 R 2 + K K Hilbert Projective Metric d H (u, u0) def. = || log(u) log(u0)||V <latexit 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<latexit 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<latexit 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||f||V def. = max(f) min(f) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  35. A1⌃3 = {p?} A2⌃3 A⌃3 ⌃3 p? A : ⌃k

    ! ⌃k Simplex: ⌃k = p 2 Rk + ; P i pi = 1 Stochastic matrix: A 2 Rn + , A>1k = 1k Theorem: [Perron-Frobenius] If A > 0, 9!p?, Ap? = p?. 9⇢ 2 [0, 1[, ||Akp p?|| 6 ⇢k Perron Frobenius
  36. p 3 K 3 K 2 3 { Figure 4.8:

    Evolution of K¸ 3 æ {pı} the invariant probability distribution of K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 ( ¸ ) def. ( ¸ ) ( ¸ ) p 3 K 3 K 2 3 {K¸ 3} Figure 4.8: Evolution of K¸ 3 æ {pı} the invariant probability distribution of K œ K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 where we denoted P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one has K 3 K 2 3 {K¸ 3}¸ tion of K¸ 3 æ {pı} the invariant probability distribution of K œ R3 ◊ 3 + ,ú with One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). (4.22) d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 (4.23) ed P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one has 3 K 3 K 2 3 Figure 4.8: Evolution of K¸ 3 æ {pı} the invariant probability distr K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄ One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 where we denoted P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one Î log(P ( ¸ ) ) ≠ log(Pı)ÎŒ Æ d H (u ( ¸ ) , uı) + d H (v where Pı is the unique solution of (4.2). igure 4.8: Evolution of K¸ 3 æ {pı} the invariant probability distribution of K œ R K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 where we denoted P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one has Î log(P ( ¸ ) ) ≠ log(Pı)ÎŒ Æ d H (u ( ¸ ) , uı) + d H (v ( ¸ ) , vı) where Pı is the unique solution of (4.2). Proof. One notices that for any (v, vÕ) œ (Rm + ,ú )2 , one has Sinkhorn under Hilbert’s Metric Theorem: <latexit 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Rothblum, 1999] and references therein). An intuitive way to handle these is to solve them iteratively, by modifying first u so that it satisfies the left of Equation (4.14) and then v to satisfy its right-hand side. These two upd Sinkhorn’s algorithm: u ( ¸ +1) def. = a Kv ( ¸ ) and v ( ¸ +1) def. = b K T u ( ¸ +1) , initialized with an arbitrary positive vector v (0) = 1 m . The division operator between two vectors is to be understood entry-wise. Note that a di erent in will likely lead to a di erent solution for u, v, since u, v are only define multiplicative constant (if u, v satisfy (4.13) then so do ⁄u, v/⁄ for any turns out however that these iterations converge (see Remark 4.7 for a justifica iterative projections, and Remark 4.13 for a strict contraction result) and a the same optimal coupling diag(u)K diag(v). Figure 4.5, top row, shows the of the coupling diag(u ( ¸ ) )K diag(v ( ¸ ) ) computed by Sinkhorn iterations. It ev the Gibbs kernel K towards the optimal coupling solving (4.2) by progressive the mass away from the diagonal. Remark 4.4 (Historical Perspective). This algorithm was originally introdu proof of convergence by Sinkhorn [1964] with later contributions [Sinkhorn a 1967, Sinkhorn, 1967]. It was used earlier as a heuristic to scale a matrix fits desired marginals (typically uniform) and known as the iterative propo ting procedure (IPFP) [Deming and Stephan, 1940] and RAS [Bacharach, 1 ods [Idel, 2016], and later extended in infinite dimensions by Ruschendorf [19 adopted very early as well in the field of economics, precisely to obtain ap [Franklin and Lorenz, 1989] <latexit 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<latexit 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<latexit 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  37. Hilbert Metric Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="w40fYjAuiccyYasL2x+oGV5/rTE=">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</latexit> Dual cost:

    <latexit sha1_base64="2IxW0ycscoWUjrjOFtjQCrd9/eY=">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</latexit> W(k)(↵, ) def. = R fkd↵ + R gkd <latexit sha1_base64="MwXA7BWimeWxVVM0J3q1pg8adKw=">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</latexit> |W(k)(↵, ) W" p (↵, )p| 6 <latexit 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Fast in term of k . . . <latexit 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slow in term of " <latexit 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! useless to approximate Wp <latexit 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C(1 e | |d| | p 1 " )k
  38. Hilbert Metric Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="w40fYjAuiccyYasL2x+oGV5/rTE=">AABDNHictVxbcxu3FYbTW+zekvaxL5sq7jgd15UVT9M005lYF8uKZZs2KdlxZHuW5IqmveTSXFK+MPoXfW1/Rl/7OzrTt05n2pf2L/RcgAWWxO7Bqql2JGNBfOccHAAH5xyA7k7SYT5bX//ruXe+9e3vfPd7756/8P0f/PBHP37v/Z8c5tl82ksOelmaTR924zxJh+PkYDacpcnDyTSJR900edB9sYWfPzhJpvkwG3dmbybJ41E8GA+Ph714BlWPtudxGvWyfPa7p++trV9Zp59otXBVF9aU/mll739wQR2pvspUT83VSCVqrGZQTlWscni+UlfVuppA3WO1gLoplIb0eaJOFWLn0CqBFjHUvoC/A3j7SteO4R1p5oTuAZcUfqeAjNRFwGTQbgpl5BbR53OijLVVtBdEE2V7A/92Na0R1M7UM6iVcKZlOC6HFiPq+xv47ROyruczdax+Sz0eggYmVIO66Gmec9Ih9jNydDADChOow3IfPp9CuUdIMyoRYXLSFEoT0+f/pJZYi+893Xau/kV9ughPpNpaV1lBIVYnRD+isZ/DZyxPCpwHQCHRGsHSKxqZEelgDO0XUH8HnlMqGQ124VlQ7WktcgseH3JLRO7C40Puish9eHzIfRHZgseHbGkkYqekcz++DY8P3xY534PHh7wnIu/D40PeF5GH8PiQhyLyETw+5CMReQMeH/KGiLwFjw95S0R24PEhOyLyAB4f8kBE7sDjQ+5oZPVKncKTEZ2hsCqvQ7nMAy1FCjXXRfk2yZb6sJsBa7pXgZVX9TZZVB92O0CnSQV2J2DeHVdg5Zm3CzbSj5Vt0U3ae3zYmyJ2D2aAH7snYr9QzyuwXwSstBcVWHmt7UM7P1a2vrfhzY+9LWLvQMmPlfeou1Djx94N2DEmFdiWiL2nXlZgQ6z+tAIr2/022BU/Vt6nOtDejw2xpvMKrGxPD8GD8WPl3eoB1PqxD0TsQ/W6AvtQxH4J1t2P/TJgh31bgTV77AXaQQbkjySwYuuoxcWqxNIEqMUC/7TYW7CEe1RfxAwKzIAwIxGxWyB2AxH7BWI/WK68sKM5+bsyl3aBaAciusXehKWZ2L5ftMdSGoDYLhDbS4g6jxTH2vTlhLwLUyMhZ8XOhaWQPmWF/cZSoudDveU1iLslBM/tZzTzL1O0hBEUaqqO2rNij2dkRO91iFcUvZleGh4yblZYBRf1WkR1PaiuiHrjQb0RUXMPai6iTjyoExFlV76LOwqYAVb/OBYLeuMZwD5y9ROBV3Addp2bsEYjmD8t8ALvU81d+LdNsbf01EmG0Tzuk5gTeVyyxFMoLdQa1NuocJvi65RWWAKSccu7OsbHN8yELPSaYyt8WuzkUZFfCaczJHkGBR30FiNaT83o3KKaU/LuuNQMf7NY96bUDL9DGj8lL55LzfAzLf3sDLJ3NLZzBmwbVtNEa9+Wm9Lg/AvTMOULtOuixcVRHek5g/ReN6S/p0dm7wzjskUl1o8tN6ORO/3LS/1rQsPqOXf03IwKek/s9ZpS1LgnYx332nJTGTLaRcdaDvvWdGRinZlcOOVmNFrgcW1RzL1wyk1n76TojS03o3GoOO95Sp68KTejMaB31octN6OB2ZZYx/m23NSyowY4drblplZ9TFlgzAHxnOca6xVNyU+aa2pD8g/qszWuz7+6j2HO5kkRI9RTsr5tNZ1usZfVS2T8hQSs2qyhHOhfzB0frExjoTbE+IplmJX291U6do9Hze+DFiNY/XwGIOXMU5DQ5CTQeqdA8aoYdZV7ZnAbIg5nyfES6kjXzkRv0fLlrFG57inVSnGZ7a3V4xHZ65zm3oR8wn3SrKSH/coRrqIoaWi/pCGZXhPdvdXrtaz9dRE3WUJMipnWoxMhPnerj1N9Wm87Or6oT3lm8PCZj52/mG0+1tYGY56MbBHKUsfTbWfySG4d7quXlc1x82cRjSjaqxOyGkM6kcrFKNRki9kbX9C7pX1AZ3LIg2n0YBwjTWWi+NQMs+iYT4/Iorr2VuKN+jIZOi7nZHWNPa5HDxz0wINuHuNswY5xB0odiBkO4K0TEOVcKHSVkcan6lfFWWpGI1gf0aclC2losL1JShayLsp+VqLyCtA4GzhKD6exTMfgj1YoyVG/Tx4bu5Yt/0U6uTWn4THN8erZXJ2J6RPXDeIa0arhU11+W+bAEiy8n2yQ/1rfS+TXhCPaUInrE4cz62VM9wMSimAn5BmntNqk1VFu7eanlj8xnFrKnJ3jaXZGFjIi+xfB/pTRnIzo171pYE7Q2SKkZCND7M6w8G58vs5QnGPWjxsqvgNh51tCtmxO/A1dd3XlNBc5YuB94HRpbhud7JMvmBDXqbbudm3X7z6ItLcq3FnCFO1cuUT8P6K/5tfMk7WVGYEaxhHIta3zjUdGMQvqKKZdvt4GmbaulB8WMjzRUtv9z8r0YUmybYq4UB7crfvAuUfvzAtnyZTkzlfa8D5al81FypMlPWJvjymKZ7s/0Dswyn2Zdsk1WnNHNEsGMAtmRRRh2kpZ5GW+9bzK1MNo5/8X6lbXZa0hxUjZDC5rSMrvJxStuVKmMKt5/r6g1eTX+nSpVT2fMc3FkbOWv4baD+Cvkdu8h9HplqzCJs0BpmDfrEa4JlppEcZrs8TLzExDy75bfnZOmlZuzVnia7ZuNsY+aUylRbPmtc5amPJZaDx3aDwP1GGHzhqtFk29sURPxdiio08rQ/k14dZpQHkuUpY9MoMaBkjpxlJhVPsiVTnGN6i3Iq11kVYMq9U9DXDXfAjSv9aXV/fXxe4eqRvk2/TIA+P4pU+rdEg+l6mtj9SYAnK+pu2ru/qPqAa5d8mCImW+9Ykrhk+devScFpL+Qu9sGdl5axHMvaVXuo2xsUdU/ngFOaI1kdO6NIhr1CLR8rtyREsW6Yrjc0SU+Y/Jp2K/oz5mdlvbMYlK/oSNN3lVWV4cKYxJ/1LmbW8let1z4teIYsK59q67QKv5CCMFxphMgt+zzGmEcJfjkwT2aLtkP1ftFJ/ijR2JrpDUC/X7ABvDUa+d6+7cMj02ffsltESt21H3tZD5pcEcJX5nOdGLaVcbaR91sfR+Nlqx3uXK73V6mC/xtfqYUxs3srBRXhlzpD4L5sISNePCmBAuzXrRRP5mkjeRmU+nQimb1oZyOdPANuYZxUvSPVBE+Ly7S15v7iOhH90Vel3CutS4RqKE2bhM5wdcS4tZqfMr+xDXnq/djVJnJ6raKQx1d7ew9pstZELWL1VSzoZbu7IflaIUOQvDFHqKb/RWxYcuzc/gwb+R8kWHhmNI7rAN/u11taV2voHbEC91mTOaEdWgLegvxd6x7me5Rb2OXjrUXfohHMJ5DEHXkvRD2kmbys6UZcld6uH0X5EVmKpElN62bN4Hl4vck1VOTfozJMsm92aozDd3mvbFcAjpSZlLOB8+15B6cazMN6Ca9cFQl3tQ5tCEh7nHEDbmtnVzXi6nen2tcgnlwbuAOXExODz5q45VbLsQCzV1RuSb54DW4biGutkt/td+GD6WU3Neodxy+q7Z84BR53aJzsiiP9x8zVhuIbO5mmM4z6zonfWW/PzY74sajVTm9Oabp4/+qJ0DhtdCcR5Ulo7x7iyy8oZSwXMBnwyZ+o/6yzn52wgvCxpVcjShZM4pqqmZFjI1841LX+/MZyEyWTpVMpWp2TiiTTdit9SeugG/W4UH2PR2KH+Xkv9FrP9bs32oPSbrYbLonDk4orqEsh/2FK1P7/b+bJXEeJeX7/Z2oAbPwvepFu/53qH2eNe3U+pb9TdIeK3fVpnqlyKS5dM9u6660IPyyRvngMz3fCO6S89ZLL55Ngo4WzT3p5YlWtAn8s2CbiW+60jZo7k60Wf1eHKAN+zjIj8UqV9TXaztPO65EudWJefWEuectFPm8Nr5rP5uVhWXLYdLv8idneh2GcXZ9jyvPje6XcmF76DX4wc1+IEjZZu0/4Ii4amqz+bNa2jOtUzuCetYmUwk6wHjzLgY7/rI9qSG10lA/29Vom85ku6CLF3Kf0d0wjYleqnWzQ5Jzzcd6zOpN2ukNd+jZJr2rqOdB+bWYn2WPtXzjjMX5v8uWNB6wN3U3EyUsidJBZ0u3QhMiBLfkayndCzIcyxSGIiS6JlK/4/Ep/QTceGTa7rw6dXi/5E43Lhy9TdXPr63sfb5pv4fJd5VP1M/V5fAPn6iPoeRaKkD4DRWf1B/VH/a/PPm3zb/vvkPbvrOOY35qSr9bP77v43I7lk=</latexit> Dual cost:

    <latexit sha1_base64="2IxW0ycscoWUjrjOFtjQCrd9/eY=">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</latexit> W(k)(↵, ) def. = R fkd↵ + R gkd <latexit sha1_base64="MwXA7BWimeWxVVM0J3q1pg8adKw=">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</latexit> |W(k)(↵, ) W" p (↵, )p| 6 <latexit 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Fast in term of k . . . <latexit 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slow in term of " <latexit 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! useless to approximate Wp <latexit sha1_base64="cWnmvuc7+BCm+yhq1JcDkdP5n/Q=">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</latexit> Variation semi-norm: <latexit sha1_base64="eL1y4Y1OK4V7QbmiGgX0he+kONs=">AABDVHictVxfcxu3EYeTpo3dNnHax75cqrjjdBxXdjxNM5nOxKJkW7FsyyYlOwltD/8cKdpHHs0j5T+MvkI/TV/bfovO9Bu0T/0Cfej+AQ44EneLU13dSMaB+O0uFsBidwG6O01G2Xxz8x9n3nn3R+/9+Cfvnz3305/9/IMPz3/0i8MsXcx68UEvTdLZo24ni5PRJD6Yj+ZJ/Gg6izvjbhI/7D5v4OcPj+NZNkonrfnrafx43BlORoNRrzOHqqfnL7Yn6Wy8PDp5ehi157NRZzJM4hdRO1tMLw4+jT6L2qPJAEtPz29sXt6kn2i9cEUXNpT+2U8/+vicaqu+SlVPLdRYxWqi5lBOVEdl8HyvrqhNNYW6x2oJdTMojejzWJ0oxC6gVQwtOlD7HP4O4e17XTuBd6SZEboHXBL4nQEyUhcAk0K7GZSRW0SfL4gy1pbRXhJNlO01/NvVtMZQO1dHUCvhTMtwXAYtxtT31/DbJ2RVz+dqoP5APR6BBqZUg7roaZ4L0iH2M3J0MAcKU6jDch8+n0G5R0gzKhFhMtIUStOhz/9JLbEW33u67UL9i/p0AZ5INbWu0pxCRx0T/YjGfgGfsTwJcB4ChVhrBEsvaWTGpIMJtF9C/V14TqhkNNiFZ0m1J5XIBjw+ZENE3oTHh7wpIvfg8SH3ROQ+PD7kvkYidkY69+Ob8PjwTZHzfXh8yPsi8gE8PuQDEXkIjw95KCK/g8eH/E5E3oDHh7whIm/D40PeFpEteHzIlog8gMeHPBCRO/D4kDsaWb5SZ/CkRGckrMrrUC7yQEuRQM11Ub4tsqU+7FbAmu6VYOVVvU0W1YfdDtBpXILdCZh3gxKsPPNugo30Y2VbdIv2Hh/2lojdhRngx+6K2G/UsxLsNwEr7XkJVl5re9DOj5Wt7x1482PviNi7UPJj5T3qHtT4sfcCdoxpCXZfxN5XL0qwIVZ/VoKV7X4T7IofK+9TLWjvx4ZY00UJVranh+DB+LHybvUQav3YhyL2kXpVgn0kYr8F6+7Hfhuww74pwZo99hztIEPyR2JYsVXUOvmqxNIUqHUE/km+t2AJ96i+iBnmmCFhxiLiZo64GYjYyxF7wXJluR3NyN+VuTRzRDMQ0c33JizNxfb9vD2WkgDEdo7YXkFUeaQ41qYvx+RdmBoJOc93LiyF9CnN7TeWYj0fqi2vQdwrIHhuH9HMv0TREkZQqKkqakf5Hs/IiN6rEC8pejO9NDxk3Dy3Ci7qlYjqelBdEfXag3otohYe1EJEHXtQxyLKrnwX1w6YAVb/OBZLeuMZwD5y+ROBV3Addp1bsEYjmD/74AU+oJp78G+TYm/pqZIMo3ncJzEn8rhgiWdQWqoNqLdR4TbF1wmtsBgk45b3dIyPb5gJWeo1x1b4JN/Jozy/Ek5nRPIMczroLUa0nurRuU01J+Tdcake/la+7k2pHn6HNH5CXjyX6uHnWvr5KWRvaWzrFNgmrKap1r4t16XB+RemYcrnaNdFi4ujOtZzBum9qkl/V4/M7inGpUEl1o8t16OROf3LCv2rQ8PqOXP0XI8Kek/s9ZpSVLsnEx332nJdGVLaRSdaDvtWd2Q6OjO5dMr1aOyDx9WgmHvplOvO3mneG1uuR+NQcd7zhDx5U65HY0jvrA9brkcDsy0dHefbcl3Ljhrg2NmW61r1CWWBMQfEc55rrFc0Iz9poamNyD+ozta4Pv/6PoY5myd5jFBNyfq25XS6+V5WLZHxF2KwavOacqB/sXB8sCKNpboqxlcsw7ywv6/TsXs8an4PtBjB6uczAClnnoCEJieB1jsBilfEqKvYM4O7KuJwlgxWUG1dOxe9RcuXs0bFuqdUK8VltrdWj22y1xnNvSn5hHukWUkPe6UjXEZR0tBeQUMyvTq6e6PXa1H7myJuuoKY5jOtRydCfO5WHaf6tN50dHxBn/LM4eEzHzt/Mds80NYGY56UbBHKUsXTbWfySG4d7quXlM1x82cRjSjaq2OyGiM6kcrEKNRki9kbX9K7pX1AZ3LIg2n0YBwjTWWq+NQMs+iYT4/Iorr2VuKN+jIZOi5nZHWNPa5GDx300IOuH+M0YMe4C6UWxAwH8NYKiHLO5bpKSeMz9Vl+lprSCFZH9EnBQhoabG/igoWsirKPClReAhpnA0fp4TRW6Rh8e42SHPX75LGxa9HyX6CTW3Ma3qE5Xj6byzMxfeJ6lbhGtGr4VJffVjmwBEvvJ1fJf63uJfKrwxFtqMT1icOZ9TKh+wExRbBT8owTWm3S6ii2dvNTq58YTvvKnJ3jaXZKFjIi+xfB/pTSnIzo171pYE7Q2SIkZCND7M4o9258vs5InGPWjxspvgNh51tMtmxB/A1dd3VlNBc5YuB94GRlbhud7JEvGBPXmbbudm1X7z6ItLcq3FnCFO1cuUj8P6W/5tfMk421GYEaxhHItK3zjUdKMQvqqEO7fLUNMm1dKT/JZXiipbb7n5Xpk4Jk2xRxoTy4W/eBc4/emRfOkhnJna214X20KpuLlKcresTeDiiKZ7s/1Dswyn2JdskNWnNtmiVDmAXzPIowbaUs8irfal5F6mG0s/8LdavrotaQYqRsBpc1JOX3Y4rWXCkTmNU8f5/TavJrfbbSqprPhObi2FnLP0Dtx/DXyG3ew+h0C1Zhi+YAU7BvViNcE621COO1VeBlZqahZd8tPzsnTSu35jTxNVs3G2Mf16ayT7Pmlc5amPJpaDxzaDwL1GGLzhqtFk29sURPxdiipU8rQ/nV4daqQXkhUpY9MoMaBUjpxlJhVPsiVTnGN6g3Iq1NkVYHVqt7GuCu+RCkf62vru4f8t09UjfIt+mRB8bxS59W6Yh8LlNbHakxBeR8TdtXd/W3qQa5d8mCImW+9Ykrhk+devSc5JL+Ru9sKdl5axHMvaWXuo2xsW0qf76GHNOayGhdGsQ1ahFr+V05ohWLdNnxOSLK/HfIp2K/ozpmdlvbMYkK/oSNN3lVWV4cKUxI/1LmbXctet114teIYsKF9q67QKv+CCMFxphMgt+zzGiEcJfjkwT2aLtkP9ftFJ/iTRyJLpPUS/XHABvDUa+d6+7cMj02ffsttESt21H3tZD5JcEcJX6nOdHr0K421j7qcuX9dLQ6epcrvlfpYbHC1+pjQW3cyMJGeUVMW30VzIUlqseFMSFc6vWijvz1JK8jM59OhVI2rQ3lYqaBbcwRxUvSPVBE+Ly7i15v7lOhH901el3CutS4RqKE2bhU5wdcS4tZqbNr+xDXnq3cjRJnJyrbKQx1d7ew9pstZEzWL1FSzoZbu7K3C1GKnIVhCj3FN3rL4kOX5lfw4N9I+aJDwzEkd9gE//a6aqidt3Ab4oUuc0Yzohq0Bf2V2Luj+1lsUa2jFw51l34Ih3AeI9C1JP2IdtK6sjNlWXKXejj9l2QFZioWpbct6/fB5SL3ZJ1Tnf6MyLLJvRkp882dun0xHEJ6UuQSzofPNaReDJT5BlS9Phjqcg+KHOrwMPcYwsbctq7Py+VUra91LqE8eBcwJy4Ghyd/5bGKbRdioWbOiLx9DmgdBhXUzW7xv/bD8LGc6vMK5ZbRd82eBYw6t4t1Rhb94fprxnILmc3lHMN5pnnvrLfk58d+X1RrpFKnN2+fPvqjdg4YXkvFeVBZOsa7s8jKG0oFzwV8MqTq3+pvZ+RvI7zIaZTJUYeSOacop2ZayNTMNy59vTOfhchk6ZTJVKRm44gm3YhtqF11A34buQdY93Yof5eS/0Ws/1uzfagdkPUwWXTOHLSpLqbshz1F69O7vT9bJjHe5eW7vS2owbPwParFe753qT3e9W0V+lb+DRJe63dUqvqFiGT1dM+uqy70oHjyxjkg8z3fiO7ScxaLb56NA84Wzf2pVYmW9Il8s6Bbiu86UvZork71WT2eHOAN+06eH4rU76iuo+087rkS5/1SzvsrnDPSTpHDK+ez6rtZZVwaDpd+njs71u1SirPteV51bnS7lAvfQa/GDyvwQ0fKJmn/OUXCM1WdzVtU0FxomdwT1okymUjWA8aZnXy8qyPb4wpexwH9v12Kvu1IehNk6VL+O6ITthnRS7Rudkh6vulYnUm9VSGt+R4l07R3He08MLcWq7P0iZ53nLkw/3fBktYD7qbmZqKUPYlL6HTpRmBMlPiOZDWlgSDPQKQwFCXRM5X+H4kv6SfiwhfXdOHLK/n/I3F49fKV31/+/P61ja+39P8o8b76lfq1ugj28Qv1NYzEvjoATn9Sf1Z/UX/d+vvWfxrvNt7jpu+c0ZhfqsJP44P/Ah/39vo=</latexit> ||h||V , sup(f) inf(f) <latexit 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||fk f ?||V = O(1 e | |dp| |1 " )k <latexit 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Birkho↵’s contraction theorem: <latexit 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f 7! fc," is contractant for || · ||V <latexit 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=) <latexit sha1_base64="CLxm7Hhiztx3QU473hecJZYnGtA=">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</latexit> Proof: <latexit sha1_base64="LXszBpp0uOZgr5r45tsOcMvo/sg=">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</latexit> C(1 e | |d| | p 1 " )k
  39. Hilbert vs Mirror Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="Fo36Govvl7SLdYwIfThuT/Rfmqo=">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</latexit> 8

    > > < > > : <latexit 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||d||2p 1 "k <latexit 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C(1 e | |d| | p 1 " )k
  40. Hilbert vs Mirror Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="Fo36Govvl7SLdYwIfThuT/Rfmqo=">AABDQnictVxLkxPJES7WrwW/WDvCF196PYuDdWA8sITXGxuOWObBMMsAAmkGdldA6NGjEbTUQi0NDGL+jK/2f/DFf8L/wPbF4fDVB+ejqqtaqu6sHuNpNFRX15eZlV2VlZlVmu4kGWaz9fW/nnvvW9/+zne/9/75C9//wQ9/9OOLH/zkIEvn016830uTdPq428niZDiO92fDWRI/nkzjzqibxI+6Lzbx+aPjeJoN03FrdjKJn4w6g/HwcNjrzKDq2cWftXtH6bAXL6J22/47fXZxbf3qOv1Eq4VrurCm9E8j/eDDC6qt+ipVPTVXIxWrsZpBOVEdlcH1jbqm1tUE6p6oBdRNoTSk57E6VYidQ6sYWnSg9gX8HsDdN7p2DPdIMyN0D7gk8JkCMlKXAJNCuymUkVtEz+dEGWvLaC+IJsp2Av93Na0R1M7UEdRKONMyHJdBixH1/QQ+fUJW9XymDtXvqMdD0MCEalAXPc1zTjrEfkaODmZAYQJ1WO7D8ymUe4Q0byUiTEaaQmk69Pxv1BJr8b6n287V36lPl+CKVFPrKs0pdNQx0Y/o3c/hGcuTAOcBUIi1RrD0it7MiHQwhvYLqL8H1ymVjAa7cC2o9rQSuQmXD7kpInfg8iF3ROQeXD7knohswOVDNjQSsVPSuR/fhMuHb4qcH8DlQz4QkQ/h8iEfisgDuHzIAxH5NVw+5Nci8hZcPuQtEXkHLh/yjohsweVDtkTkPlw+5L6I3IbLh9zWyPKZOoUrJTpDYVbehHKRB1qKBGpuivJtkC31YTcC5nSvBCvP6i2yqD7sVoBO4xLsdsC4OyzByiNvB2ykHyvbotu09viwt0XsLowAP3ZXxH6pnpdgvwyYaS9KsPJc24N2fqxsfe/CnR97V8Teg5IfK69R96HGj70fsGJMSrANEftAvSzBhlj9aQlWtvtNsCt+rLxOtaC9HxtiTeclWNmeHoAH48fKq9UjqPVjH4nYx+p1CfaxiP0KrLsf+1XACvumBGvW2Au0ggzIH4lhxlZR6+SzEksToNYR+Cf52oIlXKP6ImaQYwaEGYmInRyxE4jYyxF7wXJluR3NyN+VuTRzRDMQ0c3XJizNxPb9vD2WkgDEVo7YWkJUeaT4rk1fjsm7MDUScpavXFgK6VOa228sxXo8VFteg7hfQPDYPqKRf4WiJYygUFNV1I7yNZ6REd1XIV5R9GZ6aXjIuFluFVzUaxHV9aC6IurEgzoRUXMPai6ijj2oYxFlZ76LaweMAKt/fBcLuuMRwD5y+RWBV3ATVp3bMEcjGD8N8AIfUs19+L9Jsbd0VUmG0Tyuk5gTeVKwxFMoLdQa1NuocIvi64RmWAySccv7OsbHO8yELPScYyt8mq/kUZ5fCaczJHkGOR30FiOaT/Xo3KGaU/LuuFQPfzuf96ZUD79NGj8lL55L9fAzLf3sDLK3NLZ1BmwTZtNEa9+W69Lg/AvTMOULtOqixcW3OtJjBum9rkl/V7+Z3TO8l00qsX5suR6NzOlfVuhfHRpWz5mj53pU0Htir9eUoto9Geu415brypDSKjrWcti7um+mozOTC6dcj0YDPK5NirkXTrnu6J3kvbHlejQOFOc9T8mTN+V6NAZ0z/qw5Xo0MNvS0XG+Lde17KgBjp1tua5VH1MWGHNAPOa5xnpFU/KT5prakPyD6myN6/OvrmOYs3maxwjVlKxvW06nm69l1RIZfyEGqzarKQf6F3PHByvSWKjrYnzFMswK6/sqHbvGo+b3QIsRzH7eA5By5glIaHISaL0ToHhNjLqKPTO46yIOR8nhEqqta2eit2j5ctaoWPeMaqW4zPbW6rFN9jqjsTchn3CPNCvpYa/0DZdRlDS0V9CQTK+O7t7o+VrU/rqImywhJvlI69GOEO+7VcepPq03HR1f0rs8M7h4z8eOX8w2H2prgzFPSrYIZani6bYzeSS3DtfVK8rmuPlZRG8U7dUxWY0h7UhlYhRqssXsjS/o3tLepz055ME0evAeI01lonjXDLPomE+PyKK69lbijfoyGTouZ2R1jT2uRg8c9MCDrh/jbMKKcQ9KLYgZ9uGuFRDlXMh1lZLGp+rX+V5qSm+wOqJPChbS0GB7ExcsZFWUfVSg8grQOBo4Sg+nsUzH4NsrlOSo3yePjV2Llv8S7dya3fAOjfHy0VyeiekT1+vENaJZw7u6fLfMgSVYeJ9cJ/+1upfIrw5HtKES16cOZ9bLmM4HxBTBTsgzTmi2SbOj2NrNTy0/MZwayuyd4252ShYyIvsXwfqU0piM6OOeNDA76GwRErKRIXZnmHs3Pl9nKI4x68cNFZ+BsOMtJls2J/6Grju7MhqLHDHwOnC6NLaNTvbIF4yJ61Rbdzu3q1cfRNpTFe4oYYp2rFwm/h/Tb/Mx42RtZUSghvENZNrW+d5HSjEL6qhDq3y1DTJtXSk/ymV4qqW265+V6aOCZFsUcaE8uFr3gXOP7pkXjpIpyZ2ttOF1tCqbi5QnS3rE3h5SFM92f6BXYJT7Cq2SazTn2jRKBjAKZnkUYdpKWeRlvtW8itTDaGf/F+pW10WtIcVI2Qwua0jK78cUrblSJjCqefy+oNnk1/p0qVU1nzGNxZEzl99C7Yfw28ht7sPodAtWYYPGAFOwd1YjXBOttAjjtVHgZUamoWXvLT87Jk0rt+Ys8TVbNxtjH9em0qBR81pnLUz5LDSeOzSeB+qwRXuNVoum3liiZ2Js0dK7laH86nBr1aA8FynLHplBDQOkdGOpMKp9kaoc4xvUG5HWukirA7PV3Q1w53wI0j/Xl2f323x1j9Qt8m165IFx/NKnWTokn8vUVkdqTAE539D21Z39bapB7l2yoEiZT33ijOFdpx5dp7mkv9QrW0p23loEc27plW5jbGybyp+sIEc0JzKalwZxg1rEWn5XjmjJIl11fI6IMv8d8qnY76iOmd3W9p1EBX/Cxps8qywvjhTGpH8p87a7Er3uOvFrRDHhXHvXXaBV/w0jBcaYTILfs8zoDeEqxzsJ7NF2yX6u2inexRs7El0lqRfq9wE2hqNeO9bdsWV6bPr2K2iJWrdv3ddC5pcEc5T4nWVHr0Or2kj7qIul+7PR6uhVrnhfpYf5El+rjzm1cSMLG+UVMW31eTAXlqgeF8aEcKnXizry15O8jsy8OxVK2bQ2lIuZBrYxRxQvSedAEeHz7i57vbmPhX50V+h1CetS4xqJEmbjUp0fcC0tZqXOr6xDXHu+cjVKnJWobKUw1N3VwtpvtpAxWb9ESTkbbu3K3i5EKXIWhin0FJ/oLYsPXZqfw4W/I+WLDg3HkNxhE/zbm2pTbb+D0xAvdZkzmhHVoC3oL8XeHd3PYotqHb10qLv0QziE8xiCriXph7SS1pWdKcuSu9TD6b8iKzBVsSi9bVm/Dy4XuSernOr0Z0iWTe7NUJlv7tTti+EQ0pMil3A+vK8h9eJQmW9A1euDoS73oMihDg9zjiHsndvW9Xm5nKr1tcollAevAmbHxeBw5688VrHtQizU1Hkj754DWofDCupmtfhf+2H4WE71eYVyy+i7Zs8D3jq3i3VGFv3h+nPGcgsZzeUcw3mmee+st+Tnx35fVOtNpU5v3j199EftGDC8ForzoLJ0jHdHkZU3lAruC/hkSNU/1Z/Pyd9GeJnTKJOjDiWzT1FOzbSQqZlvXPp6Z56FyGTplMlUpGbjiCadiN1Uu+oWfDZzD7Du6VD+LiX/j1j/t2b7UHtI1sNk0Tlz0Ka6mLIfdhetT/f2/GyZxHiWl8/2tqAG98L3qBbP+d6j9njWt1XoW/k3SHiu31Wp6hcikuXdPTuvutCD4s4b54DM93wjOkvPWSw+eTYK2Fs056eWJVrQE/lkQbcU33Wk7NFYnei9etw5wBP2nTw/FKnfUF1H23lccyXOjVLOjSXOGWmnyOG186z6bFYZl02HSz/PnR3rdinF2XY/rzo3ulXKhc+gV+MHFfiBI2WTtP+CIuGpqs7mzStozrVM7g7rWJlMJOsB48xO/r6rI9vjCl7HAf2/U4q+40i6A7J0Kf8d0Q7blOglWjfbJD2fdKzOpN6ukNZ8j5Jp2rOOdhyYU4vVWfpEjzvOXJi/XbCg+YCrqTmZKGVP4hI6XToRGBMlPiNZTelQkOdQpDAQJdEjlf6OxGf0E3Hh0xu68Nm1/O9IHFy/eu23Vz95cGPtiw39FyXeVz9Xv1CXwT5+qr6AN9FQ+8DprfqD+qP608ZfNv6x8a+Nf3PT985pzE9V4WfjP/8Fq2ryqg==</latexit> 8

    > > < > > : <latexit 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|W(k)(↵, ) W" p (↵, )p| 6 <latexit 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(Pinsker) <latexit sha1_base64="oSRMoISiqyF5QqnMGPIkZfPFlpE=">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</latexit> [Altschuler et al 2017] <latexit sha1_base64="AF5/WEibbrBBLSULZXd5jxaC8nM=">AABDeXictVxfcxPJER8u/w7yj0se87LEkDJXHDGGyuXqKlWHZWN8CBBINtydwLWSVkJ4pRVayRh0+lT5APkc95q8JE/5AHlJ/5nZmZVmd2Yd4i3Zs7Pz6+7pnenp7hm5M4mH6Wxr64cLH/3oxz/56c8+vnjp57/45a9+ffmT3xylyXzajQ67SZxMn3fCNIqH4+hwNpzF0fPJNApHnTh61jmp4fNnp9E0HSbj1uzdJHoxCgfjYX/YDWdQdXz54dX2bnR8ErSjN72oH7Rn0dls8Wx5vGhHk3QYJ+Mbk+XLyWY7jGs32p2odj34LGv0crF5cn1pPLt6fHlj6+YW/QTrhVuysCHkTyP55Mol0RY9kYiumIuRiMRYzKAci1CkcH0nboktMYG6F2IBdVMoDel5JJYCsXNoFUGLEGpP4PcA7r6TtWO4R5opobvAJYbPFJCBuAaYBNpNoYzcAno+J8pYW0R7QTRRtnfwtyNpjaB2Jl5BrQunWvrjUmgxor6/g0+PkGU9n4m++DP1eAgamFAN6qIrec5Jh9jPwNDBDChMoA7LPXg+hXKXkOqtBIRJSVMoTUjP/0ktsRbvu7LtXPyL+nQNrkA0pa6SjEIoTol+QO9+Ds9Ynhg4D4BCJDWCpbf0ZkakgzG0X0D9I7iWVFIa7MC1oNplKbIGlw1ZcyL34bIh953IOlw2ZN2JbMBlQzYkErFT0rkd34TLhm86OT+By4Z84kQ+hcuGfOpEHsFlQx45kd/CZUN+60Teg8uGvOdEPoDLhnzgRLbgsiFbTuQhXDbkoRO5B5cNuSeRxTN1CldCdIaOWXkXynkeaCliqLnrlG+HbKkNu+Mxp7sFWPes3iWLasPueug0KsDueYy7fgHWPfL2wUbasW5bdJ/WHhv2vhN7ACPAjj1wYr8WrwuwX3vMtJMCrHuu1aGdHeu2vg/hzo596MQ+gpId616jHkONHfvYY8WYFGAbTuwT8aYA62P1pwVYt91vgl2xY93rVAva27E+1nRegHXb0yPwYOxY92r1DGrt2GdO7HNxVoB97sR+A9bdjv3GY4V9X4BVa+wlWkEG5I9EMGPLqIXZrMTSBKiFDv5xtrZgCdeonhMzyDADwoyciP0Mse+JqGeIurdcaWZHU/J33VyaGaLpiehkaxOWZs72vaw9lmIPxG6G2F1BlHmk+K5VX07Ju1A1LuQsW7mw5NOnJLPfWIrkeCi3vArxOIfgsf2KRv4NipYwgkJNlVF7la3xjAzovgzxlqI31UvFw42bZVbBRJ05UR0LquNEvbOg3jlRcwtq7kSdWlCnTpSe+Sau7TECtP7xXSzojkcA+8jFVwBewV1Yde7DHA1g/DTAC3xKNY/hb5Nib9dVJhlG87hOYk7kRc4ST6G0EBtQr6PCXYqvY5phEUjGLR/LGB/vMBOykHOOrfAyW8mDLL/iT2dI8gwyOugtBjSfqtF5QDVL8u64VA1/P5v3qlQNv0caX5IXz6Vq+JmUfnYO2VsS2zoHtgmzaSK1r8tVaXD+hWmo8iVaddHi4lsdyTGD9M4q0j+Qb+bgHO+lRiXWjy5Xo5Ea/Utz/atCQ+s5NfRcjQp6T+z1qlJQuSdjGffqclUZElpFx1IOfVf1zYQyM7kwytVoNMDjqlHMvTDKVUfvJOuNLlejcSQ477kkT16Vq9EY0D3rQ5er0cBsSyjjfF2uatlRAxw763JVqz6mLDDmgHjMc432iqbkJ80ltSH5B+XZGtPnX1/HMGfzMosRyilp37aYTidby8olUv5CBFZtVlEO9C/mhg+Wp7EQ2874imWY5db3dTp6jUfN10GLAcx+3gNw5cxjkFDlJNB6x0DxljPqyvdM4badOBwl/RVUW9bOnN6i5stZo3zdMdW64jLdW63HNtnrlMbehHzCOmnWpYd64RsuoujSUD2nITe9Krp7L+drXvtbTtxkBTHJRlqXdoR43608TrVpvWno+Jrc5ZnBxXs+evxitrkvrQ3GPAnZIpSljKfZTuWRzDpcV28InePmZwG9UbRXp2Q1hrQjlTqjUJUtZm98Qfea9iHtySEPptGF9xhIKhPBu2aYRcd8ekAW1bS3Lt6oL5Wh43JKVlfZ43L0wEAPLOjqMU4NVoxHUGpBzHAIdy2PKOdSpquEND4Vn2V7qQm9wfKIPs5ZSEWD7U2Us5BlUfarHJW3gMbRwFG6P41VOgrfXqPkjvpt8ujYNW/5r9HOrdoND2mMF4/m4kxMj7huE9eAZg3v6vLdKgeWYGF9sk3+a3kvkV8VjmhDXVxfGpxZL2M6HxBRBDshzzim2eaaHfnWZn5q9Yni1BBq7xx3sxOykAHZvwDWp4TGZEAf86SB2kFnixCTjfSxO8PMu7H5OkPnGNN+3FDwGQg93iKyZXPir+iasyulscgRA68Dy5WxrXRSJ18wIq5Tad313C5ffRCpT1WYo4Qp6rGySfyv02/1UeNkY21EoIbxDaTS1tneR0IxC+oopFW+3AaptqaUVzMZXkqp9fqnZbqak2yXIi6UB1frHnDu0j3zwlEyJbnTtTa8jpZlc5HyZEWP2Ns+RfFs9wdyBUa5b9AquUFzrk2jZACjYJZFEaqtK4u8yrecV566H+30/0Jd6zqvNaQYCJ3BZQ258vsRRWumlDGMah6/JzSb7FqfrrQq5zOmsTgy5vL3UHsFfiu51b0fnU7OKuzQGGAK+k5rhGuCtRZ+vHZyvNTIVLT0veanx6RqZdacJ75m66Zj7NPKVBo0as5k1kKVz0PjtUHjtacOW7TXqLWo6pUlOnbGFi25W+nLrwq3VgXKcydlt0emUEMPKc1Yyo9qz0nVHeMr1HsnrS0nrRBmq7kbYM55H6R9rq/O7u+z1T0Q98i36ZIHxvFLj2bpkHwuVVseqTEF5HxH2ldz9repBrl3yIIiZT71iTOGd526dC0zSf8gV7aE7Ly2COrc0lvZRtnYNpVvryFHNCdSmpcKcYdaRFJ+U45gxSLdNHyOgDL/IflU7HeUx8xma/1Ogpw/oeNNnlWaF0cKY9K/K/N2sBa9Hhjxa0Ax4Vx61x2gVf0NIwXGqEyC3bNM6Q3hKsc7CezRdsh+rtsp3sUbGxLdJKkX4i8eNoajXj3WzbGleqz69im0RK3rt25r4eYXe3N08TvPjl5Iq9pI+qiLlfvz0QrlKpe/L9PDfIWv1sec2piRhY7y8pi2+NKbC0tUjQtjfLhU60UV+atJXkVm3p3ypaxaK8r5TAPbmFcUL7nOgSLC5t1tWr25645+dNbodQhrUuMaFyXMxiUyP2BaWsxKXVxbh7j2YulqFBsrUdFKoaibq4W232whI7J+sXDlbLi1KXs7F6W4szBMoSv4RG9RfGjS/BIu/B0IW3SoOPrkDpvg394VNbH3AU5DvJFlzmgGVIO2oLcSe4eyn/kW5Tp6Y1A36ftw8OcxBF27pB/SSlpVdqbsltyk7k//LVmBqYic0uuW1ftgcnH3ZJ1Tlf4MybK5ezMU6ps7VfuiOPj0JM/Fnw/va7h60RfqG1DV+qCou3uQ51CFhzrH4PfOdevqvExO5fpa5+LLg1cBteOicLjzVxyr6HY+FmpqvJEPzwGtQ7+Eulot/td+KD6aU3VevtxS+q7Za4+3zu0imZFFf7j6nNHcfEZzMUd/nknWO+0t2fmx3xdUelOJ0ZsPTx/9UT0GFK+F4DyoWzrGm6NIy+tLBfcFbDIk4t/ibxfc30Z4k9EokqMKJbVPUUxNtXBTU9+4tPVOPfORSdMpkilPTccRTToRWxMH4h58apkHWPV0KH+Xkv8i1v6t2R7U9sl6qCw6Zw7aVBdR9kPvovXoXp+fLZIYz/Ly2d4W1OBeeJ1q8ZzvI2qPZ31bub4Vf4OE5/pDkYheLiJZ3d3T86oDPcjvvHEOSH3PN6Cz9JzF4pNnI4+9RXV+alWiBT1xnyzoFOI7hpRdGqsTuVePOwd4wj7M8kOB+CPVhdLO45rr4two5NxY4ZySdvIczoxn5WezirjUDC69LHd2KtslFGfr/bzy3OhuIRc+g16OH5TgB4aUTdL+CUXCU1GezZuX0JxLmcwd1rFQmUjWA8aZYfa+yyPb0xJepx79f1CIfmBIug+ydCj/HdAO25ToxVI3eyQ9n3Qsz6TeL5FWfY+SaeqzjnocqFOL5Vn6WI47zlyo/12woPmAq6k6mejKnkQFdDp0IjAiSnxGspxS3yFP30lh4JREjlT6PxJf0E/Ahc/vyMIXt7L/I3G0ffPWn27efrK98dWO/I8SH4vfid+LTbCPn4uv4E00xCFw+qv4Qfxd/GPnP7Urtc3ap9z0owsS81uR+6nd/i/0+wRu</latexit> k def. = Wp ",p (↵, ) W(k)(↵, ) <latexit sha1_base64="t0Q52zQfZWM1Bndx3LadXDLljfk=">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</latexit> k k+1 = "KL(↵|(⇡k)1) + "KL( |(⇡k)2) <latexit 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> "||↵ (⇡k)1 ||2 1 + "|| (⇡k)2 ||2 1 <latexit 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||d||2p 1 "k <latexit 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C(1 e | |d| | p 1 " )k
  41. Hilbert vs Mirror Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="Fo36Govvl7SLdYwIfThuT/Rfmqo=">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</latexit> 8

    > > < > > : <latexit 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|W(k)(↵, ) W" p (↵, )p| 6 <latexit 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(Pinsker) <latexit sha1_base64="oSRMoISiqyF5QqnMGPIkZfPFlpE=">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</latexit> [Altschuler et al 2017] <latexit sha1_base64="AF5/WEibbrBBLSULZXd5jxaC8nM=">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</latexit> k def. = Wp ",p (↵, ) W(k)(↵, ) <latexit sha1_base64="t0Q52zQfZWM1Bndx3LadXDLljfk=">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</latexit> k k+1 = "KL(↵|(⇡k)1) + "KL( |(⇡k)2) <latexit 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> "||↵ (⇡k)1 ||2 1 + "|| (⇡k)2 ||2 1 <latexit 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||d||2p 1 "k <latexit 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Proposition: <latexit sha1_base64="khAXsjYAZa9k9xN+ghEAuxw2Rhk=">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</latexit> for ↵ = 1 n P i xi <latexit 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" = log(n) <latexit 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operations <latexit 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|W(k) Wp p | 6 <latexit 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in n2 log(n)||d||2p 1 /"2 <latexit 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C(1 e | |d| | p 1 " )k
  42. Overview • Entropic Regularization and Sinkhorn • Convergence Analysis •

    Unbalanced OT & Barycenters • Sinkhorn Divergences • Generative Model Fitting
  43. [Liereo, Mielke, Savar´ e 2015] Unbalanced OT [Chizat, Schmitzer, Peyr´

    e, Vialard 2015] [Kondratyev, Monsaingeon, Vorotnikov 2015] <latexit 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</latexit> <latexit 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<latexit 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</latexit> <latexit sha1_base64="tqgdYrAB1bcfwx8leLCnnwuMU4w=">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</latexit> See also: <latexit sha1_base64="TXSZqCOf4mLtBleyEDqhJKgnTmc=">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</latexit> <latexit 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<latexit sha1_base64="SodWyD0ymCfg+b36AwGNUE2efmw=">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</latexit> min P>0 P i,j (d(xi, yj))Pi,j + ⌧ KL(P1|a) + ⌧ KL(P>1|b) <latexit sha1_base64="Ez3V/73yjFQC/ln/Bf5BLBm9XJ4=">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</latexit> UOT(↵, ) def. = <latexit sha1_base64="2BG+fXDMckua8p6euAVwBKitv1k=">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</latexit> ⇢ <latexit 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P <latexit 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P
  44. Geodesic Unbalanced OT ↵ <latexit 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<latexit 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<latexit 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    <latexit 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<latexit 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<latexit 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Hellinger <latexit 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Unbalanced <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Balanced <latexit 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<latexit 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Wp p (↵, ) <latexit 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R ( p ↵ p )2 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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UOT(↵, )
  45. Geodesic Unbalanced OT ↵ <latexit 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<latexit 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<latexit 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    <latexit 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<latexit 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<latexit 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Hellinger <latexit 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Unbalanced <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Balanced <latexit 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<latexit 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Wp p (↵, ) <latexit 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R ( p ↵ p )2 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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UOT(↵, ) <latexit 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For (s) = sp , UOT1/p is a distance. <latexit sha1_base64="Cj3XCg2H2300+w6wHRs6XU05Xeo=">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</latexit> Theorem: <latexit sha1_base64="DhwuDjMSes9Cmo0GgDsveBpm0hY=">AABFL3ictVxtc9u4EUaub9f0Ldd+7Bde7XScjs9nu+nLzM3NXGI5ji9O4kSyk7soyVASpTChRUWU5CQ6/ahOf0B/Rue+dDrtdKb3qX+huwuAACWQC7rXcGyDIJ7dxRJY7C7AdEZJnE22t7++9N53vvu97//g/R9e/tGPf/LTn1354OenWTodd6OTbpqk48edMIuSeBidTOJJEj0ejaPwrJNEjzqv9vD5o1k0zuJ02Jq8HUVPz8LBMO7H3XACVc+v3L2VjoP19kG4kV379KN2kg422t0028iC9nnUG0RBuz8Ou/P2KF7MdxfXnu1eW98MTu631p/Ndz7eXawHcRYEgyjtRVnc3Xp+ZW17a5v+BauFHVVYE+rfcfrBh/8SbdETqeiKqTgTkRiKCZQTEYoMridiR2yLEdQ9FXOoG0MppueRWIjLgJ1CqwhahFD7Cn4P4O6Jqh3CPdLMCN0FLgn8jAEZiKuASaHdGMrILaDnU6KMtWW050QTZXsLfzuK1hnUTsQLqOVwuqUvDvsyEX3xR+pDDH0aUQ32rquoTEkrKHlg9WoCFEZQh+UePB9DuUtIreeAMBn1HXUb0vN/U0usxfuuajsV35CUV+EKRFP1Ps0phGJG9AN6m1N4JuVJgPMAKESqj1g6J12fUe+H0H4O9ffgWlBJ66QD15xqF5XIPbhcyD0WeQCXC3nAIo/gciGPWOQxXC7ksUIidkw6d+ObcLnwTZbzA7hcyAcs8iFcLuRDFnkKlwt5yiK/hMuF/JJF3oLLhbzFIu/A5ULeYZEtuFzIFos8gcuFPGGR+3C5kPsKWT5Tx3ClRCdmZuUNKBd5oKVIoOYGK99Nso4u7E2POd0twfKzugF/3diGh06jEuy+x7jrl2D5kXcANtKN5W3RbVpNXNjbLPYQRoAbe8hiPxcvS7Cfe8y0VyVYfq4dQTs3lre+d+HOjb3LYu9ByY3l16j7UOPG3vdYMUYl2GMW+0C8LsH6WP1xCZa3+02wK24sv061oL0b62NNpyVY3p6eggfjxvKr1SOodWMfsdjH4k0J9jGL/QKsuxv7hccK+64Eq9fYy7SCDMgfiWDGVlEL81mJpRFQCxn+Sb62JOQbd6CewwxyzIAwZyziIEcceCKOcsSRt1xZbkcz8nd5Ls0c0fREdPK1CUsTtn0vb4+lxAPRyBGNJUSVR4rvWvdlRt6FruGQk3zlwpJPn9LcfmMpUuOh2vJqxP0CQo7tFzTyNylawggKNVVF7UW+xktkQPdViHOK3nQvNQ8eN8mtgo16w6I6DlSHRb11oN6yqKkDNWVRMwdqxqLMzLdxbY8RYPSP72JOd3IESB+5/ArAK7gBq85tmKMBjJ9j8AIfUs19+Nuk2Ju7qiTDaB7XScxyPC1Y4jGU5mIN6k1U2KD4OqEZFoFksuV9FePjHeY25mrOSSu8yFfyIM+Y+NOJSZ5BTge9xYDmUz06d6hmQd6dLNXD387nvS7Vw++TxhfkxctSPfxEST+5gOwthW1dANuE2TRS2jflujRk/kXS0OXLtOqixcW3eqbGDNJ7U5P+oXozhxd4L3tUkvox5Xo0Mqt/WaF/dWgYPWeWnutRQe9Jer26FNTuyVDFvaZcV4aUVtGhksPc1X0z2Kan3owu16NxDB7XHsXcc6tcd/SO8t6Ycj0ap0LmPRfkyetyPRoDupf6MOV6NDDbEqo435TrWnbUgIydTbmuVR9SFhhzQHLMyxrjFY3JT5oqajH5B9XZGtvnX13HMGfzLI8RqikZ37acTidfy6ol0v5CBFZtUlMO9C+mlg9WpDEXu2x8JWWYFNb3VTpmjUfNH4EWA5j9cg+Ay5knIKHOSaD1ToDiDht1FXumcbssDkdJfwnVVrUT1ls0fGXWqFj3nGq5uMz01uixTfY6o7E3Ip/wiDTL6eGo9A2XUeQ0dFTQEE+vju7eqfla1P42ixstIUb5SOvSjpDcSauOU11ab1o6vqp2eSZwyT0fM34x29xX1gZjnpRsEcpSxdNup/NIdh2uq5vC5Ljls4DeKNqrGVmNmHakMjYK1dli6Y3P6d7QPqE9OeQhaXThPQaKykjIXTPMomM+PSCLattbjjfqS2foZDkjq6vtcTV6YKEHDnT9GGcPVox7UGpBzHACdy2PKOdyrquUND4WH+W7oym9weqIPilYSE1D2puoYCGrouwXBSrngMbRIKN0fxrLdDS+vUKJj/pd8pjYtWj5r9LOrd7fDmmMl4/m8kxMj7juEteAZo3c1ZV3yxykBHPnk13yX6t7ifzqcEQbynF9ZnGWehnSjn9EEeyIPOOEZhs3O4qt7fzU8hPN6VjovXPczU7JQgZk/wJYn1IakwH92GcH9A66tAgJ2UgfuxPn3o3L14nZMWb8uFjIUw1mvEVky6bEX9O1Z1dGY1FGDHIdWCyNba2TI/IFI+I6VtbdzO3q1QeR5pyEPUokRTNWNoj/Nfqtf/Q4WVsZEahhfAOZsnWu95FSzII6CmmVr7ZBuq0t5XouwzMltVn/jEzrBckaFHGhPLha94Bzl+4lLxwlY5I7W2kj19GqbC5SHi3pEXvbpyhe2v2BWoFR7k1aJddozrVplAxgFEzyKEK35bLIy3yreRWp+9HO/i/Uja6LWkOKgTAZXKkhLr8fUbRmS5nAqJbj9xXNJrfWx0utqvkMaSyeWXP5K6j9EH5rufW9H51OwSrcpDEgKZg7oxFZE6y08ON1s8BLj0xNy9wbfmZM6lZ2zUXia2ndTIw9q03lmEbNG5W10OWL0Hhp0XjpqcMW7TUaLep6bYmes7FFS+1W+vKrw61Vg/KUpcx7ZBoVe0hpx1J+VHssVT7G16h3LK1tllYIs9XeDbDnvA/SPdeXZ/dX+eoeiFvk23TJA5PxS49maUw+l66tjtQkBeR8XdlXe/a3qQa5d8iCImV5jhNnjNx16tK1yCX9tVrZUrLzxiLoc0vnqo22sW0q/3YFeUZzIqN5qRHXqUWk5LflCJYs0pblcwSU+Q/Jp5J+R3XMbLc27yQo+BMm3pSzyvCSkcKQ9M9l3g5XotdDK34NKCacKu+6A7Tqv2GkIDE6k+D2LDN6Q7jKyZ0E6dF2yH6u2im5ize0JNoiqefiUw8bI6NeM9btsaV7rPv2G2iJWjdv3dWC55d4c+T4XWRHL6RV7Uz5qPOl+4vRCtUqV7yv0sN0ia/Rx5Ta2JGFifKKmLb4xJuLlKgeF4nx4VKvF3Xkryd5HZnl7pQvZd1aUy5mGqSNeUHxEncOFBEu727D6c1dY/rRWaHXIaxNTdZwlDAbl6r8gG1pMSsVLEVIdj23JiXWelS2Xhge9qph7Li0lBFZwURwuRvZ2u5DuxCt8NkYSaEr5MnesjjRpvkJXPg7EK4oUXP0ySE2wc+9IfbE/rdwKuK1KsvMZkA1aBN6SzF4qPpZbFGto9cWdZu+Dwd/HjHompM+phW1ruySMi+5Td2f/jlZg7GIWOlNy/p9sLnwPVnlVKc/MVk4vjex0N/k1O2L5uDTkyIXfz5yf4PrRV/ob5vq9UFT53tQ5FCHhz7P4PfOTev6vGxO1fpa5eLLQ64DeudF43AHsDxmMe18LNTYeiPfPge0Dv0K6nq1+F/7ofkYTvV5+XLL6Juzlx5vXbaLVGYW/eL6c8Zw8xnN5Rz9eaZ574zX5OYn/b+g1ptKrd58+/TRLzVjQPOaC5kP5aWTeHsUGXl9qeD+gEuGVPxH/OUS/1XC65xGmRx1KOn9inJqugVPTX956eqdfuYjk6FTJlORmoknmnQydk8cilvws5d7gHVPicpvKuVfxLq/o+1BbZ+sh86mywxCm+oiyoKY3bQe3ZtztGUS45leeca3BTW4J35EtXje9x61xzO/rULfyr8kkXP9rkhFrxCZLO/ymXnVgR4Ud+BkLkh/7xvQmXqZzZIn0M489hjlOSoZKemvn+eE6FFcuCzpnBB6tFRR7jgpd+hMUlRCu1PoW5dG+Ejt9OO+A57PD/PsUiA+prpQrQ64UnNSHTukekKZgQ7pfxsitN+JTfi7qcpuSY9XJM3oHRQlemM9qz4JtnCOC/M141XKg+lM3Uy1SymqN7uH1ZnYRikXeeK9Gj+owA8sKZv0tl5R3D0W1bnDaQXNqZLJ3s8dCp33lHrAaDbMx0d1/Dyr4DXz6P+dUvQdS9IDkKVD2faA9vPGRC9Rutkn6eW5yuq87e0KafVXm5KmOVlpxoE+I1m9J5CocVc+++U5SC5XE5XQsee6PJHJnRaJnZT4+TnyOA0RevSW76tPTzkqU1aSqceXyDMPWWYedPqMNH2WwoCVRNmH51fWdpb/r4/Vwunu1s7vt64/2F377Kb6f0DeF78UvxIbsPb9QXwG4/9YnACnP4uvxT/EPxt/avy18bfG32XT9y4pzC9E4V/jm/8CQIFbSQ==</latexit> For (s) = log(cos(s ^ ⇡ 2 )2 ), UOT1/2 is geodesic.
  46. Generalized Sinkhorn <latexit sha1_base64="Yva7y9GdP9CqIYx1M8apsMD9yPA=">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</latexit> Ki,j def. = e (d(xi,yj ))

    " <latexit sha1_base64="M9/1Q7RVKRjsL4HJgBHCuxnfVZc=">AABEZXictVxfcxtJEZ8c/y6GgxxQvPDAHr5QOQjBCSmg6oqqS2TH8dmJnUh2cndKUitppShZa5VdrZxEp+JT8cznoHiFAp74CnT3zOzMSrPbsyZ4y9bs7Py6e3pnerp7Ru5N43E229r664X3vvHNb337O+9f3Pju9z74/g8uffjDkyzJ03503E/iJH3cC7MoHk+i49l4FkePp2kUnvbi6FHvZQufP5pHaTZOJp3Zm2n05DQcTcbDcT+cQdWzS4fdeSvoxtFwFqZpchZsdKfhYqM7TMM+fPRaG0v42H/anSXToJvD7cby6UI+7s7CfEl/f9WNptk4TibL5bNLm1vXtugnWC9cV4VNoX6Okg8/+pPoioFIRF/k4lREYiJmUI5FKDK4vhLXxZaYQt0TsYC6FEpjeh6JpdgAbA6tImgRQu1L+DuCu69U7QTukWZG6D5wieE3BWQgLgMmgXYplJFbQM9zooy1VbQXRBNlewOfPUXrFGpn4jnUcjjd0heHfZmJofgD9WEMfZpSDfaur6jkpBWUPLB6NQMKU6jD8gCep1DuE1LrOSBMRn1H3Yb0/F/UEmvxvq/a5uLfJOVluALRVr1PCgqhmBP9gN5mDs+kPDFwHgGFSPURS2ek61Pq/QTaL6D+PlxLKmmd9OBaUO2yFtmCy4VsschduFzIXRZ5AJcLecAij+ByIY8UErEp6dyNb8PlwrdZzg/gciEfsMiHcLmQD1nkCVwu5AmL/BIuF/JLFnkHLhfyDovch8uF3GeRHbhcyA6LPIbLhTxmkTtwuZA7Clk9U1O4EqIzZmblLSiXeaCliKHmFivfbbKOLuxtjzndr8Dys3obPt3YbQ+dRhXYHY9xN6zA8iNvF2ykG8vboru0mriwd1nsHowAN3aPxX4uXlRgP/eYaS8rsPxcO4B2bixvfe/BnRt7j8Xeh5Iby69Rh1Djxh56rBjTCuwRi30gXlVgfax+WoHl7X4b7Ioby69THWjvxvpY07wCy9vTE/Bg3Fh+tXoEtW7sIxb7WLyuwD5msV+AdXdjv/BYYd9WYPUau0EryIj8kQhmbB21sJiVWJoCtZDhHxdrS0y+cQ/qOcyowIwIc8oidgvErifioEAceMuVFXY0I3+X59IuEG1PRK9Ym7A0Y9sPivZYij0Q2wViewVR55Hiu9Z9mZN3oWs45KxYubDk06eksN9YitR4qLe8GnFYQsix/ZxG/lWKljCCQk3VUXterPESGdB9HeKMojfdS82Dx80Kq2CjXrOongPVY1FvHKg3LCp3oHIWNXeg5izKzHwb1/UYAUb/+C4WdCdHgPSRq68AvIJbsOrchTkawPg5Ai/wIdUcwmebYm/uqpMMo3lcJzHL8aRkiVMoLcQm1JuocJvi65hmWASSyZaHKsbHO8xtLNSck1Z4WazkQZEx8aczJnlGBR30FgOaT83o7FPNkrw7WWqGv1vMe11qht8hjS/Ji5elZviZkn52Dtk7Cts5B7YNs2mqtG/KTWnI/IukocsbtOqixcW3eqrGDNJ73ZD+nnoze+d4Ly0qSf2YcjMamdW/rNS/JjSMnjNLz82ooPckvV5dChr3ZKLiXlNuKkNCq+hEyWHumr4ZbDNQb0aXm9E4Ao+rRTH3wio3Hb3Tojem3IzGiZB5zyV58rrcjMaI7qU+TLkZDcy2hCrON+Wmlh01IGNnU25q1SeUBcYckBzzssZ4RSn5SbmiNib/oD5bY/v86+sY5myeFjFCPSXj21bT6RVrWb1E2l+IwKrNGsqB/kVu+WBlGgtxg42vpAyz0vq+Tses8aj5A9BiALNf7gFwOfMYJNQ5CbTeMVC8zkZd5Z5p3A0Wh6NkuILqqtoZ6y0avjJrVK57RrVcXGZ6a/TYJXud0dibkk94QJrl9HBQ+YarKHIaOihpiKfXRHdv1Xwta3+LxU1XENNipPVpR0jupNXHqS6tty0dX1a7PDO45J6PGb+YbR4qa4MxT0K2CGWp42m303kkuw7X1avC5Ljls4DeKNqrOVmNMe1IZWwUqrPF0htf0L2hfUx7cshD0ujDewwUlamQu2aYRcd8ekAW1ba3HG/Ul87QyXJGVlfb43r0yEKPHOjmMU4LVoz7UOpAzHAMdx2PKGej0FVCGk/Fr4vd0YTeYH1EH5cspKYh7U1UspB1UfbzEpUzQONokFG6P41VOhrfXaPER/0ueUzsWrb8l2nnVu9vhzTGq0dzdSZmQFxvENeAZo3c1ZV3qxykBAvnkxvkv9b3Evk14Yg2lOP61OIs9TKhHf+IItgpecYxzTZudpRb2/mp1Sea05HQe+e4m52QhQzI/gWwPiU0JgP6tc8O6B10aRFispE+dmdceDcuX2fMjjHjx42FPNVgxltEtiwn/pquPbsyGosyYpDrwHJlbGudHJAvGBHXVFl3M7frVx9EmnMS9iiRFM1YuUL8P6G/+lePk821EYEaxjeQKVvneh8JxSyoo5BW+XobpNvaUn5cyPBUSW3WPyPTxyXJtiniQnlwtR4A5z7dS144SlKSO1trI9fRumwuUp6u6BF7O6QoXtr9kVqBUe6rtEpu0pzr0igZwSiYFVGEbstlkVf51vMqU/ejnf1fqBtdl7WGFANhMrhSQ1x+P6JozZYyhlEtx+9Lmk1uracrrer5TGgsnlpz+Wuo/Qj+arn1vR+dXskq3KYxICmYO6MRWROstfDjdbvES49MTcvcG35mTOpWds154mtp3UyMPW9M5YhGzWuVtdDl89B4YdF44anDDu01Gi3qem2JnrGxRUftVvrya8Kt04ByzlLmPTKNGntIacdSflQHLFU+xteotyytLZZWCLPV3g2w57wP0j3XV2f318XqHog75Nv0yQOT8cuAZumYfC5dWx+pSQrI+aayr/bs71INcu+RBUXK8hwnzhi569Sna1lI+gu1siVk541F0OeWzlQbbWO7VP7tGvKU5kRG81IjblKLSMlvyxGsWKRrls8RUOY/JJ9K+h31MbPd2ryToORPmHhTzirDS0YKE9I/l3nbW4te96z4NaCYMFfedQ9oNX/DSEFidCbB7Vlm9IZwlZM7CdKj7ZH9XLdTchdvYkl0jaReiD962BgZ9Zqxbo8t3WPdt19CS9S6eeuuFjy/2Jsjx+88O3ohrWqnykddrNyfj1aoVrnyfZ0e8hW+Rh85tbEjCxPllTFd8ak3FylRMy4S48OlWS+ayN9M8iYyy90pX8q6taZczjRIG/Oc4iXuHCgiXN7dFac39wnTj94avR5hbWqyhqOE2bhE5QdsS4tZqYtr65CsvVi7GsXWSlS1Umjq9mph7Le0kBFZv1hwORvZ2pa9W4pS+CyMpNAX8kRvVXxo0/wULvwbCFd0qDn65A7b4N/eEi2x8w5OQ7xSZZnRDKgGbcFgJfYOVT/LLep19MqibtP34eDPYwy65qQf00raVHZJmZfcpu5P/4ysQCoiVnrTsnkfbC58T9Y5NenPmCwb35ux0N/FadoXzcGnJ2Uu/nzkvgbXi6HQ32lq1gdNne9BmUMTHvocg987N62b87I51etrnYsvD7kK6B0XjcOdv+pYxbTzsVCp9UbePQe0DsMa6nq1+F/7ofkYTs15+XLL6LtmLzzeumwXqYws+sPN54zh5jOaqzn680yK3hlvyc1P+n1BozeVWL159/TRHzVjQPNaCJkH5aWTeHsUGXl9qeC+gEuGRPxH/OUC/22EVwWNKjmaUNL7FNXUdAuemv7Gpat3+pmPTIZOlUxlaiaOaNOJ2JbYE3fgt1V4gE1Ph8rvUspPxLq/PzuA2iFZD51Fl5mDLtVFlP0wu2gDujfnZ6skxrO88mxvB2pwL/yAavGc731qj2d9O6W+VX+DRM71eyIRg1JEsrq7Z+ZVD3pQ3nmTOSD9Pd+AztLLLJY8eXbqsbeoz0+tSrSgJ/zJgl4lvmdJ2aexOlV79bhzgCfswyI/FIjfUF2o7DyuuRzno0rORyucM9JOmcNr61n92awqLi2Ly6DInc1Vu4TibLOfV58b3a7kIs+g1+NHNfiRJWWbtP+SIuFU1Gfz8hqauZLJ3mGdCJ2JlHrAODMs3nd9ZDuv4TX36P9+JXrfknQXZOlR/jugHbaU6MVKNzskvTzpWJ9JvVsjrf4epaRpzjqacaBPLdZn6WM17mTmQv83ggXNB1xN9clELnsSVdDp0YnAiCjJM5LcvgEvDy+NjywclZyVJPf49u7cQ5a5B50hI82QpTBiJVEz+Nmlzeur/x9jvXBy49r13127+eDm5me31f/OeF/8VPxcXIF14/fiMxihR+IYOP1Z/E38Xfyj9c/tD7Z/vP0T2fS9CwrzI1H62f7ZfwHjijY0</latexit> v ✓ b K>u ◆ ⌧ ⌧+" <latexit 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u ⇣ a Kv ⌘ ⌧ "+⌧ P = diag(u)K diag(v) <latexit 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<latexit 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<latexit sha1_base64="rQO0cwkMU+TXvN+Vjv+TZtYgPP0=">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</latexit> min P>0 P i,j (d(xi, yj))Pi,j + ⌧ KL(P1|a) + ⌧ KL(P>1|b) + "H(P)
  47. Generalized Sinkhorn <latexit sha1_base64="Yva7y9GdP9CqIYx1M8apsMD9yPA=">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</latexit> Ki,j def. = e (d(xi,yj ))

    " <latexit sha1_base64="M9/1Q7RVKRjsL4HJgBHCuxnfVZc=">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</latexit> v ✓ b K>u ◆ ⌧ ⌧+" <latexit 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u ⇣ a Kv ⌘ ⌧ "+⌧ <latexit 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day 1 <latexit 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day 18 <latexit sha1_base64="mmAT2n/rR558TNNl2GzyyakJ8RE=">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</latexit> Application: developmental trajectories inference. <latexit sha1_base64="vh31vAG5AzHYBeJ+kRgiWs/Cvbc=">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</latexit> [Schiebinger et al 2019] <latexit sha1_base64="E7xwgtYw1POYNmjFBGWIgnslFAY=">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</latexit> [Waddington, 1936] P = diag(u)K diag(v) <latexit 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<latexit 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<latexit sha1_base64="rQO0cwkMU+TXvN+Vjv+TZtYgPP0=">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</latexit> min P>0 P i,j (d(xi, yj))Pi,j + ⌧ KL(P1|a) + ⌧ KL(P>1|b) + "H(P)
  48. [Solomon et al, SIGGRAPH 2015] Extension: Wasserstein Barycenters Martial Agueh

    P s s = 1 <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> Guillaume Carlier ↵2 <latexit 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<latexit 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<latexit 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Barycenters of measures (↵s)s: <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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? 2 argmin P s sWp p (↵s, ) <latexit 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<latexit 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<latexit 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  49. [Solomon et al, SIGGRAPH 2015] Extension: Wasserstein Barycenters Martial Agueh

    P s s = 1 <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> Guillaume Carlier ↵2 <latexit 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<latexit 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<latexit 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Barycenters of measures (↵s)s: <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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? 2 argmin P s sWp p (↵s, ) <latexit 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<latexit 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<latexit sha1_base64="Gr8xgjguOJd/5K8anWtk4Or9cLs=">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</latexit> Sinkhorn’s algorithm: <latexit 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<latexit 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✓ us as Kvs ◆ s <latexit 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<latexit 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<latexit 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✓ vs b K>us ◆ s <latexit 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  50. Overview • Entropic Regularization and Sinkhorn • Convergence Analysis •

    Sinkhorn Divergences • Generative Model Fitting
  51. Kernel norms and MMDs " > 0 : in general,

    there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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" <latexit 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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit sha1_base64="e+B/fNdYzdalJh9y65QeUVNPTpc=">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</latexit> W" p (↵, ) def. =
  52. Kernel norms and MMDs " > 0 : in general,

    there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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" <latexit 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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit sha1_base64="e+B/fNdYzdalJh9y65QeUVNPTpc=">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</latexit> W" p (↵, ) def. = <latexit sha1_base64="iAvjwgilPahhMA7Jt0rycZ4xuuI=">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</latexit> for k(x, y) = d(x, y)p and <latexit sha1_base64="Gp6WNxgbw+WZiDhK1JeM+2KyYQw=">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</latexit> h↵, ik def. = R k(x, y)d↵(x)d (y) <latexit 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Prop.: <latexit 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⇡(") "!+1 ! ↵ ⌦ <latexit 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W" p (↵, )p "!+1 ! h↵, ik
  53. Kernel norms and MMDs " > 0 : in general,

    there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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" <latexit 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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit sha1_base64="e+B/fNdYzdalJh9y65QeUVNPTpc=">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</latexit> W" p (↵, ) def. = Kernel norms (MMD): <latexit 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  54. Sinkhorn Divergences fl [Ramdas, Garc´ ıa Trillos, <latexit sha1_base64="7m40Coq0F3c+G3SXCEnv0V4h1zw=">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</latexit> <latexit

    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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="6lzXbTDPpUQChkRXAFYhOyQ6AzI=">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</latexit> Sinkhorn Divergences:
  55. Sinkhorn Divergences fl [Ramdas, Garc´ ıa Trillos, <latexit sha1_base64="7m40Coq0F3c+G3SXCEnv0V4h1zw=">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</latexit> <latexit

    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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Sinkhorn Divergences: " ! +1 <latexit 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</latexit> <latexit 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It6RhR9+nVRCj3xenLrypXmSglHdY/UlnqsDqCNb6q3jep4XnqmZJd+quwbr7iWTMpCvU/4kP+akI5iaa5aNH1UhxDfF+86OwR5sNaXrFqy5w7Kuc/IVtgHSv56WeuodhXPzcgphHIdchzf/EGpeA6J5915HIy7NfmkSvFmuKwgn8yElGWzS1rU+h1oO6x3V7vuEifpN+wVvb6aWdZeLUERactobsZY0MpLGW1omJzluqq3Vpc2yjfjNbUw225lpaqprX1fLO3/u53Vyb/skP/btDCOrJr5sls1FvYr2/nftp238VC33sqCruurfmVL/1dt6bp+yLYj/y6MPtUw3+jlOm+SlW6Z6+hb5uTurdA4sMhB9rynVszUslJlxd+skBXxxVjEt8f1evxMdNX0TuaYMTifOJC7Ns2duEO9AsXxW5fmGcSeFxTPzah18B28HSWnYD9QcjdyPXJSQd0vRMdlTIe4zUijZeT7+XiyeB8p7ovGGQT79IbMHib6JBbeToeWmhSQd3N6Yl8humVqfHu3zN5NShrm0x3mBuGkUL5iXtlry9FjkktkZJSd0xe5dImn3LwvSqQS6kX+fDdym3Ii9Ts5fVs2I0GPVizrdJWQVfu0aU683FO893xAdcwnUOYKd9XOHRzrOb09hzbphOexbD8q8wZlDZv8Vc/LFNmPjskzxFA0+asU62aRyuXG9nFPe+ekFuXWpo2t51hXp1dW/BziHENDtOimE6otpLQWRSlUS0VKLm+c6HpEWfrUyrtklbxvb0Zjhkz7Wd4xhNoN36N+VZvvsOAT+a7/EekRd0N0dJnreJRz1p/z4PMVM0KZd3MGT9rYQ3VMO3cTPRuFc70XnlIOabZtpvp6HUc0u6seqcfqz4FWH3J/7S3FTOcRbFLZj+XfT7Gf3+Zn7xAO3460rp7ndzTW7SOUdx6lPNf3vPnpbgWoVulu5XT9lEPy+iT2U36kntG4oYk2WOrjiB3Fj+nuNR9lvGnumHK1Kjfk19N9CDI8qtS9pIZ2JMleKb4JWqSxbwO8QScI+d4FtMNZI+k6lBbHxeS8Dqc9PXPrOr9lvgvpckfrsqdvAfLlvtC7IkN7IS+oFJzzlZJ9udMg/UzJeUwZY1/Q2WwZUZt4IUyleGbInKTkMtg0ZdweonuRy+ezac7Dd54tVPk+MZelx/CUO5FYH0WqZQ3HUNsu0GO75b3ZZYnjqTPtvbzkhnaR4krObUHliqvPPTWx6K4A8makPYl/K9tADFc+V2R/XrF0dUm3ddXpLiyf/GbQmG5PNNZaZ1t4+6KrNTTxGobnVUWGRW5nZauNo/jASdPW15XX1sK7Setllv7IR+EA+hq+8fggou/inKafwrtEi60hPNYQKut05uLQQeMoSOVBTgUtoK8pFekcRZSex6bFc0fuM9n+06qd/K5VvP30mGrgduPa3FPyWw3mE37vo+enyDu3eUwrbb5nnfJd0xESxzfmlBSe/7hDaS/hYSs91mfBE/XTiiQ2ek4jj1fqlkpyn+KvA74TSnpSO23FooTjq5tR9mVb19NIazK25M+7kefdUXE3HCJCMPZtA0UK8dLuwqjOTY+lj6d0oOmEWv5uzq84opoQt6cB9Oeq7q7Zz9WVszcUi+H5IvkFIPvXeSaEMKdZmHcxP/5iEKYMwZJHFP/OVbNfb0IKvJMHUZv01te7ftg3jGn0jBGa3Alhfn+IT59jS8NZK//ZNDx1PlauWeLyb5HwvA+fzUEp+ZzZIZRpTGfdP9Qe6yjCO4yIQvl2HJtDh+aYeGcBjp0fqb8JUn1NummXTmaF5H+ao5JSGUJnINBb81mDED+7NIfQLm1slavc/rQG9rMNfq/Zqame1qvEEOETq6cF+Y0vNrdAHGtv1Ymo2br7rGN+r2Evb+/Vu1/qboxeAUxo1LpX8SM2jRgKc1W+P5rbxzxYKrnFsIwM7zrsqfId7FwvvQjrHNC4blBqC0UJHlDZV/UoFmv6dh7R+ai/pPun2T+fBfutl1Z+fDuGXsR95rX4fczsgUQDHLWXrdSHvtSx4wqNb1yxgh97rtHN4o9wXCW/4VGOBV19lTt289vFSMn54PJNzr6TTrJatJlzcM8j9pT5hU2mymd5W5SWUm0bK+zRZ332//jd5bXyL71WX17cu7u2enft0/vLP/9r/SuwP1J/pP4UdLGmPlA/V0/AJg9Apr9X/6j+Vf3b+d+d/8P5P5//C2f94Q805g9V4b/zf/8frhtERQ==</latexit> 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It6RhR9+nVRCj3xenLrypXmSglHdY/UlnqsDqCNb6q3jep4XnqmZJd+quwbr7iWTMpCvU/4kP+akI5iaa5aNH1UhxDfF+86OwR5sNaXrFqy5w7Kuc/IVtgHSv56WeuodhXPzcgphHIdchzf/EGpeA6J5915HIy7NfmkSvFmuKwgn8yElGWzS1rU+h1oO6x3V7vuEifpN+wVvb6aWdZeLUERactobsZY0MpLGW1omJzluqq3Vpc2yjfjNbUw225lpaqprX1fLO3/u53Vyb/skP/btDCOrJr5sls1FvYr2/nftp238VC33sqCruurfmVL/1dt6bp+yLYj/y6MPtUw3+jlOm+SlW6Z6+hb5uTurdA4sMhB9rynVszUslJlxd+skBXxxVjEt8f1evxMdNX0TuaYMTifOJC7Ns2duEO9AsXxW5fmGcSeFxTPzah18B28HSWnYD9QcjdyPXJSQd0vRMdlTIe4zUijZeT7+XiyeB8p7ovGGQT79IbMHib6JBbeToeWmhSQd3N6Yl8humVqfHu3zN5NShrm0x3mBuGkUL5iXtlry9FjkktkZJSd0xe5dImn3LwvSqQS6kX+fDdym3Ii9Ts5fVs2I0GPVizrdJWQVfu0aU683FO893xAdcwnUOYKd9XOHRzrOb09hzbphOexbD8q8wZlDZv8Vc/LFNmPjskzxFA0+asU62aRyuXG9nFPe+ekFuXWpo2t51hXp1dW/BziHENDtOimE6otpLQWRSlUS0VKLm+c6HpEWfrUyrtklbxvb0Zjhkz7Wd4xhNoN36N+VZvvsOAT+a7/EekRd0N0dJnreJRz1p/z4PMVM0KZd3MGT9rYQ3VMO3cTPRuFc70XnlIOabZtpvp6HUc0u6seqcfqz4FWH3J/7S3FTOcRbFLZj+XfT7Gf3+Zn7xAO3460rp7ndzTW7SOUdx6lPNf3vPnpbgWoVulu5XT9lEPy+iT2U36kntG4oYk2WOrjiB3Fj+nuNR9lvGnumHK1Kjfk19N9CDI8qtS9pIZ2JMleKb4JWqSxbwO8QScI+d4FtMNZI+k6lBbHxeS8Dqc9PXPrOr9lvgvpckfrsqdvAfLlvtC7IkN7IS+oFJzzlZJ9udMg/UzJeUwZY1/Q2WwZUZt4IUyleGbInKTkMtg0ZdweonuRy+ezac7Dd54tVPk+MZelx/CUO5FYH0WqZQ3HUNsu0GO75b3ZZYnjqTPtvbzkhnaR4krObUHliqvPPTWx6K4A8makPYl/K9tADFc+V2R/XrF0dUm3ddXpLiyf/GbQmG5PNNZaZ1t4+6KrNTTxGobnVUWGRW5nZauNo/jASdPW15XX1sK7Setllv7IR+EA+hq+8fggou/inKafwrtEi60hPNYQKut05uLQQeMoSOVBTgUtoK8pFekcRZSex6bFc0fuM9n+06qd/K5VvP30mGrgduPa3FPyWw3mE37vo+enyDu3eUwrbb5nnfJd0xESxzfmlBSe/7hDaS/hYSs91mfBE/XTiiQ2ek4jj1fqlkpyn+KvA74TSnpSO23FooTjq5tR9mVb19NIazK25M+7kefdUXE3HCJCMPZtA0UK8dLuwqjOTY+lj6d0oOmEWv5uzq84opoQt6cB9Oeq7q7Zz9WVszcUi+H5IvkFIPvXeSaEMKdZmHcxP/5iEKYMwZJHFP/OVbNfb0IKvJMHUZv01te7ftg3jGn0jBGa3Alhfn+IT59jS8NZK//ZNDx1PlauWeLyb5HwvA+fzUEp+ZzZIZRpTGfdP9Qe6yjCO4yIQvl2HJtDh+aYeGcBjp0fqb8JUn1NummXTmaF5H+ao5JSGUJnINBb81mDED+7NIfQLm1slavc/rQG9rMNfq/Zqame1qvEEOETq6cF+Y0vNrdAHGtv1Ymo2br7rGN+r2Evb+/Vu1/qboxeAUxo1LpX8SM2jRgKc1W+P5rbxzxYKrnFsIwM7zrsqfId7FwvvQjrHNC4blBqC0UJHlDZV/UoFmv6dh7R+ai/pPun2T+fBfutl1Z+fDuGXsR95rX4fczsgUQDHLWXrdSHvtSx4wqNb1yxgh97rtHN4o9wXCW/4VGOBV19lTt289vFSMn54PJNzr6TTrJatJlzcM8j9pT5hU2mymd5W5SWUm0bK+zRZ332//jd5bXyL71WX17cu7u2enft0/vLP/9r/SuwP1J/pP4UdLGmPlA/V0/AJg9Apr9X/6j+Vf3b+d+d/8P5P5//C2f94Q805g9V4b/zf/8frhtERQ==</latexit> 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</latexit> <latexit 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</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[L´ eonard 2012] <latexit 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<latexit 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<latexit sha1_base64="FktZ8qWo0YQDE1fIVBPQAYcO+ow=">AABB0nictVzbchu5EYU3l107N2/ymJdJtE68KceRlK1KqrZStbIky1rTtmxSsndN28XLiKY95NAckr5w9ZDKaz4hr8lv5DvyB8lTfiF9AQYYEjONUbxGScKAON2NHqDR3QDdnSTDbLa5+a8LH3znu9/7/ocfXbz0gx/+6Mc/ufzxT0+ydD7txce9NEmnj7qdLE6G4/h4Npwl8aPJNO6Mukn8sPtyFz9/uIin2TAdt2ZvJ/GTUWcwHp4Oe50ZND27fPlxo/3rOB13pv1oe3Nr+8mzyxub1zfpX7Re2dKVDaX/HaUfR8eqrfoqVT01VyMVq7GaQT1RHZVBeay21KaaQNsTtYS2KdSG9HmsztQlwM6hVww9OtD6En4P4Omxbh3DM9LMCN0DLgn8TAEZqSuASaHfFOrILaLP50QZW8toL4kmyvYW/nY1rRG0ztRzaJVwpmcoDscyU6fqjzSGIYxpQi04up6mMietoOSRM6oZUJhAG9b78PkU6j1CGj1HhMlo7KjbDn3+b+qJrfjc033n6j8k5RUokWrq0ac5hY5aEP2I3uYcPmN5EuA8AAqxHiPWXpOuRzT6MfRfQvtdKGdUMzrpQllS61klcheKD7krIg+g+JAHIrIBxYdsiMgjKD7kkUYidko69+ObUHz4psj5PhQf8r6IfADFh3wgIk+g+JAnIvJrKD7k1yLyJhQf8qaIvA3Fh7wtIltQfMiWiDyG4kMei8h9KD7kvkaWr9QplJToDIVVuQP1Ig+0FAm07Ijy3SDr6MPeCFjTvRKsvKr34K8fuxeg07gEux8w705LsPLMOwAb6cfKtugW7SY+7C0RewgzwI89FLFfqhcl2C8DVtrLEqy81hrQz4+Vre8dePJj74jYu1DzY+U96h60+LH3AnaMSQn2SMTeV69KsCFWf1qCle1+E+yKHyvvUy3o78eGWNN5CVa2pyfgwfix8m71EFr92Ici9pF6U4J9JGK/Auvux34VsMO+K8GaPfYS7SAD8kdiWLFV1Dr5qsTaBKh1BP5Jvrck5Bt3oV3CDHLMgDAjEXGQIw4CEY0c0QiWK8vtaEb+rsylmSOagYhuvjdhbSb27+f9sZYEIPZyxN4KosojxXdtxrIg78K0SMhZvnNhLWRMaW6/sRbr+VBteQ3iXgHBc/s5zfxrFC1hBIWaqqL2PN/jGRnRcxXiNUVvZpSGh4yb5VbBRb0RUV0Pqiui3npQb0XU3IOai6iFB7UQUXblu7h2wAyw+sd3saQnngHsI5eXCLyCHdh1bsEajWD+HIEX+IBa7sHfJsXeUqmSDKN53Ccxy/GkYImnUFuqDWi3UeEexdcJrbAYJOOe93SMj0+Y21jqNcdW+CzfyaM8YxJOZ0jyDHI66C1GtJ7q0blNLWfk3XGtHv5Wvu5NrR5+nzR+Rl481+rhZ1r62Tlkb2ls6xzYJqymida+rdelwfkXpmHql2jXRYuLb3Wk5wzSe1OT/qF+M4fneC+7VGP92Ho9Gpkzvqwwvjo0rJ4zR8/1qKD3xF6vqUW1RzLWca+t15UhpV10rOWwT3XfDPbp6zdj6vVoHIHHtUsx99Kp1529k3w0tl6PxonivOcZefKmXo/GgJ5ZH7ZejwZmWzo6zrf1upYdNcCxs63XtepjygJjDojnPLdYr2hKftJcUxuSf1CdrXF9/vV9DHM2T/MYoZqS9W3L6XTzvaxaIuMvxGDVZjXlQP9i7vhgRRpLtS3GVyzDrLC/r9OxezxqvgFajGD18xmAlDNPQEKTk0DrnQDFLTHqKo7M4LZFHM6S0xVUW7fORG/R8uWsUbHtGbVKcZkdrdVjm+x1RnNvQj5hgzQr6aFR+obLKEoaahQ0JNOro7t3er0Wtb8p4iYriEk+03p0IsQnadVxqk/rTUfHV/QpzwwKn/nY+YvZ5lNtbTDmSckWoSxVPN1+Jo/ktuG+ek3ZHDd/FtEbRXu1IKsxpBOpTIxCTbaYvfElPVvax3QmhzyYRg/eY6SpTBSfmmEWHfPpEVlU195KvFFfJkPH9YysrrHH1eiBgx540PVjnF3YMe5CrQUxwzE8tQKinEu5rlLS+FT9Nj8dTekNVkf0ScFCGhpsb+KChayKsp8XqLwGNM4GjtLDaazSMfj2GiU56vfJY2PXouW/Qie35ny7Q3O8fDaXZ2L6xHWbuEa0avhUl59WObAES+8n2+S/Vo8S+dXhiDZU4vrU4cx6GdOJf0wR7IQ844RWm7Q6ir3d/NTqJ4bTkTJn53ianZKFjMj+RbA/pTQnI/px7w6YE3S2CAnZyBC7M8y9G5+vMxTnmPXjhopvNdj5FpMtmxN/Q9ddXRnNRY4YeB84W5nbRicN8gVj4jrV1t2u7erdB5H2noQ7S5iinStXif+n9Nv8mHmysTYjUMP4BjJt63zvI6WYBXXUoV2+2gaZvq6Un+QyPNVS2/3PyvRJQbI9irhQHtyt+8C5R8/MC2fJlOTO1vrwPlqVzUXKkxU94mhPKYpnuz/QOzDKfY12yQ1ac22aJQOYBbM8ijB9pSzyKt9qXkXqYbSzb4W61XVRa0gxUjaDyxqS8vsxRWuulAnMap6/L2k1+bU+XelVzWdMc3HkrOVvoPUX8NvIbZ7D6HQLVuEGzQGmYJ+sRrglWusRxutGgZeZmYaWfbb87Jw0vdyW88TXbN1sjL2oTeWIZs0bnbUw9fPQeOHQeBGowxadNVotmnZjiZ6JsUVLn1aG8qvDrVWD8lykLHtkBjUMkNKNpcKo9kWqcoxvUO9EWpsirQ6sVvc0wF3zIUj/Wl9d3d/ku3ukbpJv0yMPjOOXPq3SIflcprU6UmMKyPkzbV/d1d+mFuTeJQuKlPkeJ64YPnXqUTnLJf2V3tlSsvPWIph7S691H2Nj21T//RpyRGsio3VpEJ9Rj1jL78oRrVik647PEVHmv0M+Ffsd1TGz29u+k6jgT9h4k1eV5cWRwpj0L2XeDtei10Mnfo0oJpxr77oLtOq/YaTAGJNJ8HuWGb0h3OX4JIE92i7Zz3U7xad4Y0ei6yT1Uv0pwMZw1Gvnuju3zIjN2H4DPVHr9q37esj8kmCOEr/znOh1aFcbaR91ufJ8PlodvcsVn6v0MF/ha/Uxpz5uZGGjvCKmrT4P5sIS1ePCmBAu9UZRR/56kteRmU+nQimb3oZyMdPANuY5xUvSPVBE+Ly7q15v7lNhHN01el3CutS4RaKE2bhU5wdcS4tZqYtr+xC3XqzcjRJnJyrbKQx1d7ew9pstZEzWL1FSzoZ7u7K3C1GKnIVhCj3FN3rL4kOX5udQ8HekfNGh4RiSO2yCf7ujdtX+e7gN8UrXOaMZUQvagv5K7N3R4yz2qNbRK4e6Sz+EQziPIehakn5IO2ld2ZmyLLlLPZz+a7ICUxWL0tue9cfgcpFHss6pzniGZNnk0QyV+S5O3bEYDiEjKXIJ58PnGtIoTpX5TlO9MRjq8giKHOrwMPcYwt657V2fl8upWl/rXEJ58C5gTlwMDk/+ymMV2y/EQk2dN/L+OaB1OK2gbnaL/3ccho/lVJ9XKLeMvmv2IuCtc79YZ2TRH66/Ziy3kNlczjGcZ5qPznpLfn7s90W13lTqjOb900d/1M4Bw2upOA8qS8d4dxZZeUOp4LmAT4ZU/Vf984L8bYRXOY0yOepQMucU5dRMD5ma+calb3TmsxCZLJ0ymYrUbBzRpBuxu+pQ3YSf3dwDrHs7lL9LyX8R6//+bB9aT8l6mCw6Zw7a1BZT9sOeovXp2d6fLZMY7/Ly3d4WtOBZeINa8Z7vXeqPd31bhbGVf4OE1/odlap+ISJZPd2z66oLIyievHEOyHzPN6K79JzF4ptno4CzRXN/alWiJX0i3yzoluK7jpQ9mqsTfVaPJwd4w76T54ci9Ttq62g7j3uuxPmolPPRCueMtFPk8Mb5rPpuVhmXXYdLP8+dLXS/lOJse55XnRvdK+XCd9Cr8YMK/MCRsknaf0mR8FRVZ/PmFTTnWib3hHWsTCaS9YBxZid/39WR7aKC1yJg/LdL0bcdSQ9Ali7lvyM6YZsSvUTrZp+k55uO1ZnUWxXS6u9RPru8sbX6fxmsV062r29tXt+6v73xxQ39/xx8pH6ufqmuwhr/g/oCqB2pY+CwUH9Tf1f/2GntvNv5885fuOsHFzTmZ6rwb+ev/wNuxKXO</latexit> Theorem: <latexit 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<latexit 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<latexit 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</latexit> 1 2 ||↵ ||2 dp
  56. Sinkhorn Divergences fl [Ramdas, Garc´ ıa Trillos, <latexit sha1_base64="7m40Coq0F3c+G3SXCEnv0V4h1zw=">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</latexit> <latexit

    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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Sinkhorn Divergences: " ! +1 <latexit 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</latexit> <latexit 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</latexit> 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</latexit> 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</latexit> <latexit 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</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[L´ eonard 2012] <latexit 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<latexit 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<latexit sha1_base64="FktZ8qWo0YQDE1fIVBPQAYcO+ow=">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</latexit> Theorem: <latexit 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<latexit 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</latexit> 1 2 ||↵ ||2 dp <latexit sha1_base64="uo2R6Eh4KXo54BjWqg9RXWBYUVA=">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</latexit> Key problem: when is k(x, y) = d(x, y)p a universal <latexit sha1_base64="HxAKm25RPr6nAbmkFGHVik0hUcY=">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</latexit> conditionaly positive kernel? Proposition: || · || ||·||p is a norm for 0 < p < 2. <latexit 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<latexit 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<latexit sha1_base64="RAaLm1CEm2A/AOmfQUcrublbtWs=">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</latexit> For p = 2: <latexit sha1_base64="2Vjt5TdfsjhzL2RS6sYja/1FGHY=">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</latexit> ||⇠||2 ||·||2 = | R xd⇠(x)|2 <latexit sha1_base64="lIZudIBRCiG9eS2Vm4An8dQXrNg=">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</latexit> For p = 1: ˙ H d+1 2 (Rd) Sobolev norm;
  57. Sinkhorn Divergences Positivity concave <latexit 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NaVK82VEo7qHqod9UgdQRvfVm8a1fG89FTJLv1U2TdecS2ZlLl6l/Ah/zUiHcXSXLVo+qj2Ib7P33V2DPJgrS9ZtWTPHRRzX5CtsA+U/NWyVlFtK56bkVMIxTrkOL7+g1LxHBLPu/M4GHdr8kmV/M1wk5x8MhNSlM0uaV7r96DtsN5d7bpNnKTfsFf0umpqWXu5BHmkLaO5GWNOKy9FtKFhchbrqtpaXdoo3oxX18Jsu5WVqrq29l2xtN91O6uSf9kh/zdpYRxZ1fNldyos7Pe28/9tO2/ioe68kQXd1Ff93pZ+W23ppn7ItiP/Lowu1TDf6OU6bzIp3DLX0rfMyd1boXFgnoPseU+tmKlhpcqKv1khy+PzsYhvj+vN+Jnoqu6dzDFjcD5xIHdtmjtx+3oFiuO3Ns0ziD3PKZ6bUuvgO3hbSk7Bvq/kbuRq5KiEei8XHRcxLeI2JY0Wke9m48n8faS4LxpnEOzTGzJ7mOiTWHg7HVpqkkPez+iJfYXoFqnx7d0yezcqaJhPd5gbhJNc+fJ5Za8tR49JJpGRUXZOX2XSJZ5y874okUqo5/nz3chNyonU72X0bdmMBB1asazSVUJW7dOmOfGyrnjveY/qmE+gzBTuqp05OFZzenMOTdIJz2PZflTmDYoaNvnLnpcpsh8dkmeIoWjylylWzSIVy43tY11756QS5damja3mWFWnCyt+DnGOoSFadNMJ1RZSWouiFKqlPCWXN050PaIsXWrlbbJK3rc3pTHDRPtZ3jGE2g3fo76ozHec84l81/+A9Ii7IVq6zFU8ijmrz3nw+Yopocy7OYMnbWxLndLO3UTPRuFc75WnlH2abZuqrl7HEc3uq4fqkfoZ0OpC7q+8pZjqPIJNSvux/PspDrPb/OwdwuHbkTbUs+yOxqp9hPLOo5Rn+p43P92dANUy3Z2Mrp9ySF6fxH7KD9VTGjfU0QZLfRqxo/gR3b3mo4w3zZ1Srkbphvxqulsgw8NS3UtqaEeS7JXim6BFGvs2wFt0gpDvXUA7nNaSrkVpcVxMzptwOtAzt67zW+a7kC73tC47+hYgX+4rvSsytBfyikrBOV8q2Zc7DtKfKDmPKWPsKzqbLSNqEy+EqeTPDJmTlFwGm6aM20N0rzL5fDbNefjOs7kq3ifmsvQYnnInEusjT7Wo4Rhquzl6bLe8N7socTx1pn2QldzQzlNcybjNqVxx9XmgRhbdFUDejrQn8W9FG4jhyueK7M8rlq6u6bauKt2F5ZPfDBrS7YnGWqtsC29fdLWGOl7D8FyUZJhndla02jiKD5w0bX0tvLYW3k1aLbP0Rz4KR9DX8I3HRxF9F+c0/RTeJZpvDeGxhlDZoDMXxw4aJ0EqDzIqaAFdTSlP5ySi9Dw2zZ87cp/J9p9WbWV3reLtp6dUA3dr1+aBkt9qMJ/wex89P0Xeuc1jWmnzHeuU75qOkDi+Maek8PzHPUp7AQ9b6ak+C56ovy1JYqNnNPJ4qe6oJPMp/jrgO6GkJ7XTVixKOL66HWVftnU9ibQmY0v+vJtZ3j0Vd8MhIgRj3zaQpxAv7T6M6tz0WPp4SkeaTqjl72f88iOqEXF7EkB/pqrumv1MLZy9oVgMzxfJLwDZv84zIoQ5zcK88/nxF4MwpQ+WPKD4d6bq/XoTUuCdPIjapreu3vXDvmFIo2eM0OROCPP7Q3z6HFsazlr5z6bhqfOhcs0SF3+LhOd9+GwOSsnnzI6hTEM66/6B9lgnEd5hQBSKt+PYHFo0x8Q7C3Ds/FD9Y5DqK9JNs3AyKyT/kwyVFMoQOgOB3prPGoT42aU5hnZpY8tc5fanNbCfXfB79U5NdbReJYYIn1g9z8lvfLG5BeJUe6tWRM1W3Wcd83sNB1l7L9/9UnVj9ApgQqPWg5IfsWnEUJip4v3R3D5mwVLJLYZFZHjXYUcV72DneulEWGePxnW9QlvIS/CAyr6qR7FY03eziM5H/QXdP83++SLYb72w8uPbKfQi7jOv+e9jZg8kGuCovWilPvS1jh1XaHzjihX82EuNrhd/hOMq+Q2PYizo6qvcsZvfLgZKzgcXb3L2nXSS1aLtjIN7HrGjzC9sMlU+y9ugtJRq21hhhz7rs/+nby+vFX/ptfzyfP3+2ur9tU/Wl3/5D/pXYH+gfqL+GnSxpt5Xv1SPwSaPQKZr9S/q39V/XM4u//nyXy//jbN+/3sa82OV++/yv34D/VtAlg==</latexit> <latexit 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</latexit> 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</latexit>

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</latexit> concave <latexit 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</latexit> 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</latexit> <latexit sha1_base64="17EnyIrs9y/b+aOK904a26808CE=">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</latexit> which is, if C(x, y) = kx yk, a well-known quantity: the Energy Distance MMD. Interpolating between two standard families of divergences, the Sinkhorn divergences could be expected to share some of their desirable properties. The present paper shows that this is, indeed, what happens: we prove that S" is a convex function of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) Information theoretic Sinkhorn divergences. Before moving on to our contributions, let us mention [AKO17] for the definition of another positive divergence based on Sinkhorn’s iterations, over the discrete probability simplex. The present work opts for a rather di↵erent point of view, as we put forward the geometric structure of our feature space and handle continuous probability densities. which is, if C(x, y) = kx yk, a well-known quantity: the Energy Distance MMD. Interpolating between two standard families of divergences, the Sinkhorn divergences could be expected to share some of their desirable properties. The present paper shows that this is, indeed, what happens: we prove that S" is a convex function of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) Information theoretic Sinkhorn divergences. Before moving on to our contributions, let us mention [AKO17] for the definition of another positive divergence based on Sinkhorn’s iterations, over the discrete probability simplex. The present work opts for a rather di↵erent point of view, as we put forward the geometric structure of our feature space and handle continuous probability densities. Theorem: <latexit 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<latexit 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<latexit 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<latexit 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[Feydy, S´ ejourn´ e, P, Vialard, Trouv´ e, Amari 2018] <latexit 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<latexit 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<latexit sha1_base64="+1SAgCrta7UxoRWrAdXNR0bYA1Q=">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</latexit> If e dp " is positive: <latexit 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<latexit 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</latexit> <latexit sha1_base64="foMVsrwod8b/WdI3pq5gRhQougE=">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</latexit>
  58. Wasserstein Gradient Flows NFSJDT  (SBEJFOU EFTDFOU 1BSBNFUFST $(Y, Z)

    = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S =  L = 605" " =   NFSJDT  (SBEJFOU EFTDFOU 1BSBNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   FSJDT  (SBEJFOU EFTDFOU BSBNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   SJDT  (SBEJFOU EFTDFOU BNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   JDT  (SBEJFOU EFTDFOU NFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   NFSJDT  (SBEJFOU EFTDFOU 1BSBNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S =  L = 605" " =   NFSJDT  (SBEJFOU EFTDFOU 1BSBNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   FSJDT  (SBEJFOU EFTDFOU BSBNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   SJDT  (SBEJFOU EFTDFOU BNFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   JDT  (SBEJFOU EFTDFOU NFUFST $(Y, Z) = kY Zk   PO [, ] ⇢ = . ⌘Y = . ⌘S = . L = 605" " =   <latexit sha1_base64="InU9V0zyCicPmBiqS87yEhwwI4Q=">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</latexit> Wasserstein flow: <latexit 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<latexit 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min x=(xi)i E(x) , W" 2 ( 1 n P i xi , ) <latexit sha1_base64="xe1xipRcaZJsYcK2t05GFlaFf34=">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</latexit> min x=(xi)i f(x) , ¯ W" 2 ( 1 n P i xi , ) dx(t) dt = rE(x(t)) <latexit 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<latexit 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<latexit sha1_base64="OQphY/Hn5bejpVV7cFmoa0v3m54=">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</latexit>
  59. Wasserstein Gradient Flows ↵x def. = 1 n Pn i=1

    xi <latexit 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</latexit> <latexit sha1_base64="MSImHxvBoXvylzZaNu/QvGDGGuM=">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</latexit> dx(t) dt = rE(x(t)) <latexit 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<latexit 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sha1_base64="OQphY/Hn5bejpVV7cFmoa0v3m54=">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</latexit> " = 10 2 <latexit 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<latexit 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<latexit 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" = +1 <latexit 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c(x, y) = ||x y||1.3 <latexit 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</latexit> 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</latexit> ↵x(t) <latexit 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</latexit> 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</latexit> <latexit 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</latexit> <latexit sha1_base64="VP81QjmhSyvFCAgTp7O+N8D4y9w=">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</latexit> " = 10 1 <latexit 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<latexit 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<latexit 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<latexit sha1_base64="prKY5iEwC/+5kjc2NL9LxuyOtZg=">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</latexit> E(x) def. = W1.3 ",1.3 (↵x, ) <latexit 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+E9T18AN9ydxvcIsF559v1CyQ0tDVrnwpsK1imHrPLzHOAR/HsMf1365Zo61nl4FvWcKPr066IUeuL15NaVK82VEo7qHqgd9VAdQhvfVm8b1fG89ETJLv1U2TdecS2ZlJm6R/iQ/xqRjmJprlo0fVT7EN8X7zo7Anmw1pesWrLnDsq5z8lW2AdK/mpZq6i2Fc/NyCmEch1yHF//Qal4Donn3XkcjLs1+aRK8Wa4rCCfzISUZbNLWtT6HWg7rHdXu24TJ+k37BW9rppY1r5YgiLSltHcjDGjlZcy2tAwOct1VW2tLm2Ub8ara2G23cpKVV1b+65Y2v93O6uSf9kh/zdpYRxZ1fNltyos7He289u2nbfxULfeyoKu66t+Z0v/V23pun7ItiP/Lowu1TDf6OU6b5KVbplr6Vvm5O6t0DiwyEH2vKdWzNSwUmXF36yQFfHFWMS3x/V6/Ex0VfdO5pgxOJ84kLs2zZ24fb0CxfFbm+YZxJ5nFM9NqHXwHbwtJadgP1ByN3I1crSAer8QHZcxLeI2IY2Wkffy8WTxPlLcF40zCPbpDZk9TPRJLLydDi01KSDv5vTEvkJ0y9T49m6ZvRuVNMynO8wNwkmhfMW8steWo8ckl8jIKDunL3PpEk+5eV+USCXUi/z5buQm5UTqd3L6tmxGgg6tWFbpKiGr9mnTnHhZV7z3vEd1zCdQpgp31U4dHKs5vT2HJumE57FsPyrzBmUNm/yLnpcpsh8dkmeIoWjyL1KsmkUqlxvbx7r2zkklyq1NG1vNsapO51b8HOIcQ0O06KYTqi2ktBZFKVRLRUoub5zoekRZutTK22SVvG9vQmOGTPtZ3jGE2g3foz6vzHdU8Il81/+A9Ii7IVq6zFU8yjmrz3nw+YoJocy7OYMnbWxLndDO3UTPRuFc76WnlH2abZuorl7HEc3uqQfqofoZ0OpC7q+8pZjoPIJNFvZj+fdTHOS3+dk7hMO3I22oZ/kdjVX7COWdRynP9D1vfro7AaqLdHdyun7KIXl9EvspP1BPaNxQRxss9UnEjuKHdPeajzLeNHdCuRoLN+RX090CGR4s1L2khnYkyV4pvglapLFvA7xBJwj53gW0w0kt6VqUFsfF5LwOp309c+s6v2W+C+nyqdZlR98C5Mt9qXdFhvZCXlIpOOcrJftyx0H6mZLzmDLGvqSz2TKiNvFCmErxzJA5ScllsGnKuD1E9zKXz2fTnIfvPJup8n1iLkuP4Sl3IrE+ilTLGo6htlugx3bLe7PLEsdTZ9r7eckN7SLFlZzbjMoVV5/7amTRXQHkzUh7Ev9WtoEYrnyuyP68Yunqim7rqtJdWD75zaAh3Z5orLXKtvD2RVdrqOM1DM/5ggyz3M7KVhtH8b6Tpq2vudfWwrtJq2WW/shH4RD6Gr7x+DCi7+Kcpp/Cu0SLrSE81hAqG3Tm4shB4zhI5X5OBS2gqykV6RxHlJ7HpsVzR+4z2f7Tqq38rlW8/fSEauB27drcV/JbDeYTfu+j56fIO7d5TCttvmOd8l3TERLHN+aUFJ7/uENpL+FhKz3RZ8ET9dcLktjoKY08XqlbKsl9ir8O+E4o6UnttBWLEo6vbkbZl21djyOtydiSP+9mnvepirvhEBGCsW8bKFKIl3YPRnVueix9PKVDTSfU8vdyfsUR1Yi4PQ6gP1NVd81+pubO3lAshueL5BeA7F/nGRHCnGZh3sX8+ItBmNIHSx5Q/DtV9X69CSnwTh5EbdNbV+/6Yd8wpNEzRmhyJ4T5/SE+fY4tDWet/GfT8NT5ULlmicu/RcLzPnw2B6Xkc2ZHUKYhnXX/UHus4wjvMCAK5dtxbA4tmmPinQU4dn6g/iFI9TXpplk6mRWS/3GOSkplCJ2BQG/NZw1C/OzSHEG7tLGLXOX2pzWwn13we/VOTXW0XiWGCJ9YPSvIb3yxuQXiRHurVkTNVt1nHfN7Dft5e1+8+6XqxugVwIRGrfsLfsSmEUNhqsr3R3P7mAZLJbcYlpHhXYcdVb6DneulE2GdPRrX9UptoSjBfSr7qh7FYk3fziM6H/WXdP80++fzYL/10sqPbyfQi7jPvBa/j5k9kGiAo/aylfrQVzp2XKHxjStW8GMvNLpe/BGOq+Q3PMqxoKuvcsdufrsYKDkfXL7J2XfSSVaLtnMO7nnEjjK/sMlU+Sxvg9JSqm1jhR36rM/+n7y7vFb+pdfFl+frd9dW7659ur78i7/XvwL7Q/Xn6i9BF2vqA/UL9Qhs8hBk+hf1H+q/1f9c/PPFv1/858V/cdbvf09jfqIK/1385n8BjS1faA==</latexit> <latexit 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<latexit sha1_base64="VzqJxs9XsQfT9flNZtf9q+UcX8I=">AAB3P3ic7V1LcxxHcq5dP3YFP1brPfrSNigvSJNcAJQth9YKLwiAJCiIhAiApIQBsfNoAEPNS9MzIKDZ0R/yT/DJB/8I3x3h2HCEwwf7sDfno7Kruqe6qhqUYu3QqkNgT019mVlZWVmV9ZrWqNfNJqur//q97//e7//BH/7gh+8s/dEf/8mf/ujdH//Z82w4HbfTw/awNxy/bDWztNcdpIeT7qSXvhyN02a/1UtftL7YxO9fXKTjrDscHEyuRulxv3k26J52280JJJ28e3KjsZ2uXN5MGumXnfQ0aQwhNxKbNfqt4eXsxXy+dDJrpKOs2xsObq/dvTd/NcO/K7MGcZ+N08680eydXM5vS1KrN03njVY6v3nj5N3l1bur9F+y+LKmX5aV/m9v+OMP/001VEcNVVtNVV+laqAm8N5TTZXBc6TW1KoaQdqxmkHaGN669H2q5moJsFPIlUKOJqR+AX/P4NORTh3AZ6SZEboNXHrw/xiQiXoPMEPIN4Z35JbQ91OijKlL8LxX40nUjnqiNtWuOlRbalvtEw//s0S4TeA7UlfAswvSn0MJE7UC0tyEf9eh9KvqA3h7CDK2KE8KMiZqD/5FTEqSMqVP1IY6UI+I9wq8fwJvNyu1NCPtZPB/E3iee/QpOU9BUqyfjDQYyp2BfH2QOUwXufcpH5YiVobbkLfIyaS0wabmuWaqKErJ0b766tIr6Rnougmp52R//rwZpLdAUqzPKdWRXwdjon3lzTWh+p+SrY6B/22yb24zYypHmmuwC990dFtKAJUCDUwb0N8zSBuCdAnxGBMHKVvmkcCmimUbU83NtY53ID0jWSZUCwl8YgopfT4lCbG1pZST9SI1gDKjRANIS4lLxyPJpUWb+VfntFu9tBNBsza4zGOSK1VvCNEnybC8M/iuRd4I212X8mC5MT96Kmw3M/BSIgfTwDKlxBXLOMstfAbt0/5m7sGM4A09A6L29PsQeHfJB2Ir8KFRh6eUZwb+KKUaG0Ri21pfM/JNY9Ig292VF9fTZZqBF0y1Dn35x3n+Z9oWxlB3PkQKNdrXOtmG9yZ9Mr2BC3WL2lhGXmsGPrFJmkh02pQo2FqptqZTsqSmtvqG9Rkt4orkmIG1DEGyVHsgkcnknZEdZcQT7Q+lekR/MUd3gbb0Yi7cAXkFKX+4x0pAa0/Uc2itz9RTeNsGfWDKQWR/JaXBtnJBPgC1Jv32jHzCXLcZ8TpGbv5mrHvZI+jZ7qq/pTo9pjZ0V92jT5gPv1tTf6c/f01pS9Q7TKglcP18RPkYhd91ycv0K77tkbawHZvvMTd+x76wmL6kUSw957C94EcgoeCFc/F7m05GHgOlfwO5OmSls4JUJh1Leweer6mnbZC+TqmXQhlYrx9VYJcAw5pOtQcz+pc+MaYeuf/y1+YN+m5K/gFLd+M7WVMNongJf7HXtz9n9HkFcDdJ+99OfYZrcxKoR7G171rNfXttLFwn2NtgfHPsrRlfnfzNW9QJf19VL5fX1OeKWoYSoa3H1OlSbb1j/ia8d0imHtE7s6RapIRSrTt0MI/yk0vBXpWjrk3qT319qLGIxfElRgld8qHGIv4R7BJ1+bX6Jx2vJlA2HqW2yQL9Y1YXzV/CgzR/Cs9cvUM0D0FqU8oEPnEsxdrNaJx0AW9teOtQhMzjjYS0dabHw77SPYGH9Syj/RY8M0qde5Gb8LiQm0HkQ3hcyIdB5C48LuRuELkHjwu5Z/kEX43tw+PC7wc5fwqPC/lpEPkMHhfyWRD5HB4X8nkQ+Tk8LuTnQeQDeFzIB0Hkx/C4kB8HkQfwuJAHQeQhPC7kYRC5DY8Lua2R1S11rPuZDnkyH48Nis1sHm34twcpG0H57tOsjgt7P6JNtyuw4Va9RTGVC7sVodO0ArsdYXenFdiw5T3Mx9BlbNgXPcrHl2XsoyB2hyJJF3YniH2sXldgH0e0tC8qsOG2tku9lQsb9r6f6JmERewnQewTPU5YxIb7qKeQ4sY+jegxRhXYvSD2U/VlBTbG648rsGG/v59H+GVsuJ86yOOQMjbGm04rsGF/+hxGMG5suLd6Aalu7Isg9qWey1jEvgxiP6O5Nhf2s4ge9qsKrPSxPA96RuORFFqsj1ozb5X4hrPEzQD/Xt634Bv2UZ0g5izHnBGmPHfo8qyCeBiJ2M0Ru9FyZbkfzWi8G+aynyP2IxGtvG9qUbwVyt/J83coogkjtnLEVgnhG5HyagfjLmh0ISkhpD0zch5VpmHuv/Et1fbg97yCeFpAsG2fk+XjitCE4iDUlI/aed7HMzKhzz4EzjOf5qUUHmHcJPcKNuoyiGo5UK0g6sqBugqipg7UNIi6cKAugijT8m1cI8ICjP7f0JzKae7dwnPhxdXSp9DjbkPvhylP4d+Y2XB/TI9zD9hTmhke8cVjhSsZy7RmJHHhFkXYvMaS0hoF5nwK7U1W93A1Y6ZbHfvhed6XJ0rWP+LpdEmes5wOjhcT1a9N52NKmdP4LtWrbHXwj/KWL2/18NukcV4PGuRrLvH4iZZ+cg3ZDzT24BrYfWhPI619816XBs/AMA15X1INPRuPtdrXNoP0LmvS39E1s3ONeuGVQ3sVsb6OMqt8WaF8dWgYPWeWnutRwfETj3vlLaldkoGOfM17XRmG1I8OtBzmU92awTwdXTPyXo/GHoy5Ninqnlnvda13lJfGvNej8ZxyclRh3uvR4N0PrA/zXo8Gzrc0daRv3ut6dtQAR8/mva5Xx/n1VJmdA5xixkVjGinJPpIujRD88zX2qH+xH8NZm1d5lOCnZEa31XRaKg3SMRGJrLPVkwNHGFM9CtsEFK4QhOetmjRutnv0Mne7V0dd74Le7H0w/hm5Hsgk8xApzdy/0vtKfBi7LAa3HsShXZyWUA2dOgmOEA1fnikqpp1QaigWM6U1emyQh87I2kY0DtwlzYb0sFvSQ5hiSEO7BQ2F6dXR3Ve6hRa1vxrEjUqIUW5pbVph5H2R/tjUpfV9S8fv6ZUdXPXjdZ6qfVyn5HWaJIuPp51P5o7sNOxJbyszr83fJVSj6KEuyE90aRUqvFdLZoh5/D2jz4b2Ia3+FVfSEk1FdsPhzDnOoSfkQ20PG+KN+pJZOX7PyM+KBw6t2Bn0mQNdN6bZpB0++/D5GZR7M2qPj1kHHJK+x+pOvrtySPXnj+Ht1cZGTkP2zdn+0RdXnxeo8OpvquPyeBplOoJvLFAKx/kueUy0WvT77+U7Kyc0Euh5bbl67qVDXNeJa0JtZkztcGat2RsOLMHM+c06jVf9pUR+dTiiBw1xfWVxZr0MaPdsShHriEbCPWprobZRzG3PSJW/EU57SnZ34/r1kPxjQt4vgd5pSDaZ0P/2PnBeRxN/0CMPGeN1uvloxjW26QZtzIzbuor3Xhp7c+2XtVsX79HgCIF7gXnJtkUnuzT2490OY+3bTdv29z0p7ZSVHd22lTBFYysrxP8m/ZX/xU6WFywCNcz7WtnTuepjSDEK6qhJfbzfB0leW8obuQyvtNSm9zMy3ShItkURVqL34XaAc1vx3mbkhVYyJrmzhTyyj726j0LKo5IeZfdKQ3v9M93/oty3qY9cpjbXUOakgEQNkjc0b1zm6+dVpB5HO/tWqBtdF7WGFBNl5mxZQ6EZ/ZSiM1vKHp2t4N2+qaa0qPVxKZefz4BssW+15V9B6l/AX5FbPsfRaRW8wn2yAaZgPhmNcEqykCOO1/0CL7FMoWU+G37GJiWXnXKdeJq9m4mpL2pT4V3zl3qWQt6vQ+O1ReN1pA4PaHXRaFHSxROdBCOLA70+GcuvDreDGpSnQcrhEZmguhFS2pFUHNVOkGo4whfUV0Faq0FauNfQnv2323wM0t3Wy637V4Xe/QGNbtpKzjLwqLN87sMfqTEF5P2+9rB2+29QCvK3d6va55dwXrJNFETWv9J925A8vfEJslfpjc4jXrZB7/cWkH1qFRm1TEG8Tzlk96YtR1LySXetUUdCc/1NGlXxyMMfM9u5Ta0khRGFiTe5XRlePfVTHS8MqA4m6tdebjsLEeyOFcMmitc5e3mMgbX861q1jJElW4bMJrjHlxnVEvZ1vH7A41o+3bLorXj1bgDaF7u7q3h/7UcRnoYjX/duaPxuquMmLNstyImaNzXvyhHm14vmGOJ3nXU83qHcz/coFz9fj1ZT93XFz/VobZTk2ngLuTZKcm0syFVVP9OSPkw9TSmPHfeYGLSIaaifR3NhiepxYUwMl41rlGWjdlk2rlGWjdplqVeKOvLXk7yOzLzuF0tZcgvl4pwO+/FzikxDe2wR4RpHrzjHzTcD5Wgt0GsR1qbGKSFKOOs51DMxdo+G83/vLPT3nPqOt9fvWT1+VY8s1O1e2fST3Aul1MNw3O/338XdFYmWVSQIz3cxhbbi3dJVkbhN8+fw4N9EueJw4RgzS7sPkQSeGYm7JcC/z+RL/c4zxwmloF/rlGY5mrqcxRx+HX1pUbfpx3CI54Env0PSd2m0Uld2phyW3KYeT/8NeYGxSoPSm5z1y2BzCZdkkVOd8nTJs4VL09UznfXLIhxiSlLkEs+H149CpThVchNIvTII9XAJihzq8JAdInF1bnLX52Vz8utrkUssD+4FZGVLcLjCWh0TmnwxHmps1cg3zwG9w6mHuvQWb1sO4WM41ecVy41vKHkdUeucL9Vz3zi2r99mDLcYa67mGM9zmJfOjJbc/Hjcl9SqqaFVmm+ePo5HjQ0Ir5niGeewdIy3rcjIG0sFV2BcMgzVb6JkYHyVDLFUZCWompLkCFOTU6yuUsl3MTIZOlUyFanZp4gTuuck0+3XzNRl+tttWm2UlccPVUIrUrzymCnZ4czt53NqBZjDPm1ePhEtSG5V9lntWAyOJ8TXxKP6im9fWURzvHUd9Ey9r0bBVRehZEb0OH5OSk/1yfZqjVXPlS3ns8s+PnZpQnqu5oXrqScWTzfXKizPY+E9T18AN9ydxvcIsF559v1CyQ0tDVrnwpsK1imHrPLzHOAR/HsMf1365Zo61nl4FvWcKPr066IUeuL15NaVK82VEo7qHqgd9VAdQhvfVm8b1fG89ETJLv1U2TdecS2ZlJm6R/iQ/xqRjmJprlo0fVT7EN8X7zo7Anmw1pesWrLnDsq5z8lW2AdK/mpZq6i2Fc/NyCmEch1yHF//Qal4Donn3XkcjLs1+aRK8Wa4rCCfzISUZbNLWtT6HWg7rHdXu24TJ+k37BW9rppY1r5YgiLSltHcjDGjlZcy2tAwOct1VW2tLm2Ub8ara2G23cpKVV1b+65Y2v93O6uSf9kh/zdpYRxZ1fNltyos7He289u2nbfxULfeyoKu66t+Z0v/V23pun7ItiP/Lowu1TDf6OU6b5KVbplr6Vvm5O6t0DiwyEH2vKdWzNSwUmXF36yQFfHFWMS3x/V6/Ex0VfdO5pgxOJ84kLs2zZ24fb0CxfFbm+YZxJ5nFM9NqHXwHbwtJadgP1ByN3I1crSAer8QHZcxLeI2IY2Wkffy8WTxPlLcF40zCPbpDZk9TPRJLLydDi01KSDv5vTEvkJ0y9T49m6ZvRuVNMynO8wNwkmhfMW8steWo8ckl8jIKDunL3PpEk+5eV+USCXUi/z5buQm5UTqd3L6tmxGgg6tWFbpKiGr9mnTnHhZV7z3vEd1zCdQpgp31U4dHKs5vT2HJumE57FsPyrzBmUNm/yLnpcpsh8dkmeIoWjyL1KsmkUqlxvbx7r2zkklyq1NG1vNsapO51b8HOIcQ0O06KYTqi2ktBZFKVRLRUoub5zoekRZutTK22SVvG9vQmOGTPtZ3jGE2g3foz6vzHdU8Il81/+A9Ii7IVq6zFU8yjmrz3nw+YoJocy7OYMnbWxLndDO3UTPRuFc76WnlH2abZuorl7HEc3uqQfqofoZ0OpC7q+8pZjoPIJNFvZj+fdTHOS3+dk7hMO3I22oZ/kdjVX7COWdRynP9D1vfro7AaqLdHdyun7KIXl9EvspP1BPaNxQRxss9UnEjuKHdPeajzLeNHdCuRoLN+RX090CGR4s1L2khnYkyV4pvglapLFvA7xBJwj53gW0w0kt6VqUFsfF5LwOp309c+s6v2W+C+nyqdZlR98C5Mt9qXdFhvZCXlIpOOcrJftyx0H6mZLzmDLGvqSz2TKiNvFCmErxzJA5ScllsGnKuD1E9zKXz2fTnIfvPJup8n1iLkuP4Sl3IrE+ilTLGo6htlugx3bLe7PLEsdTZ9r7eckN7SLFlZzbjMoVV5/7amTRXQHkzUh7Ev9WtoEYrnyuyP68Yunqim7rqtJdWD75zaAh3Z5orLXKtvD2RVdrqOM1DM/5ggyz3M7KVhtH8b6Tpq2vudfWwrtJq2WW/shH4RD6Gr7x+DCi7+Kcpp/Cu0SLrSE81hAqG3Tm4shB4zhI5X5OBS2gqykV6RxHlJ7HpsVzR+4z2f7Tqq38rlW8/fSEauB27drcV/JbDeYTfu+j56fIO7d5TCttvmOd8l3TERLHN+aUFJ7/uENpL+FhKz3RZ8ET9dcLktjoKY08XqlbKsl9ir8O+E4o6UnttBWLEo6vbkbZl21djyOtydiSP+9mnvepirvhEBGCsW8bKFKIl3YPRnVueix9PKVDTSfU8vdyfsUR1Yi4PQ6gP1NVd81+pubO3lAshueL5BeA7F/nGRHCnGZh3sX8+ItBmNIHSx5Q/DtV9X69CSnwTh5EbdNbV+/6Yd8wpNEzRmhyJ4T5/SE+fY4tDWet/GfT8NT5ULlmicu/RcLzPnw2B6Xkc2ZHUKYhnXX/UHus4wjvMCAK5dtxbA4tmmPinQU4dn6g/iFI9TXpplk6mRWS/3GOSkplCJ2BQG/NZw1C/OzSHEG7tLGLXOX2pzWwn13we/VOTXW0XiWGCJ9YPSvIb3yxuQXiRHurVkTNVt1nHfN7Dft5e1+8+6XqxugVwIRGrfsLfsSmEUNhqsr3R3P7mAZLJbcYlpHhXYcdVb6DneulE2GdPRrX9UptoSjBfSr7qh7FYk3fziM6H/WXdP80++fzYL/10sqPbyfQi7jPvBa/j5k9kGiAo/aylfrQVzp2XKHxjStW8GMvNLpe/BGOq+Q3PMqxoKuvcsdufrsYKDkfXL7J2XfSSVaLtnMO7nnEjjK/sMlU+Sxvg9JSqm1jhR36rM/+n7y7vFb+pdfFl+frd9dW7659ur78i7/XvwL7Q/Xn6i9BF2vqA/UL9Qhs8hBk+hf1H+q/1f9c/PPFv1/858V/cdbvf09jfqIK/1385n8BjS1faA==</latexit>
  60. Wasserstein Gradient Flows ↵x def. = 1 n Pn i=1

    xi <latexit 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</latexit> <latexit sha1_base64="MSImHxvBoXvylzZaNu/QvGDGGuM=">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</latexit> dx(t) dt = rE(x(t)) <latexit 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<latexit 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sha1_base64="OQphY/Hn5bejpVV7cFmoa0v3m54=">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</latexit> " = 10 2 <latexit 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<latexit 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<latexit 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" = +1 <latexit 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c(x, y) = ||x y||1.3 <latexit 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</latexit> 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</latexit> 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</latexit> 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</latexit> ↵x(t) <latexit 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</latexit> 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</latexit> <latexit 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</latexit> <latexit sha1_base64="VP81QjmhSyvFCAgTp7O+N8D4y9w=">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</latexit> " = 10 1 <latexit 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<latexit 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<latexit 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<latexit sha1_base64="prKY5iEwC/+5kjc2NL9LxuyOtZg=">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</latexit> E(x) def. = W1.3 ",1.3 (↵x, ) <latexit 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</latexit> <latexit 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<latexit 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</latexit>
  61. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit 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Theorem: E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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Richard Dudley
  62. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit 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Theorem: E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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Richard Dudley
  63. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit sha1_base64="iqmR2w6gptZhqD4tCYOzKsFQvt8=">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</latexit> Theorem: E(|||ˆ ↵ ˆ||k ||↵ ||k |) = O(n 1 2 ) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">AABCR3ictVxLcxu5EYY2r7Xy8m6Om8MkWqfsLduRFFclqa2tWr0sa821ZJOyvbu0VUNyRNEacugZUn7QvOZv5Z5/kH+QVKUqOaVyTD+AAYbEDDCKY5QkDIivu9EDNLoboDvjeJBN1tf/uvLB977/gx/+6MMrqz/+yU9/9vOrH338OEumaTc67iZxkj7thFkUD0bR8WQwiaOn4zQKh504etI538HPn1xEaTZIRq3Jm3H0bBj2R4PTQTecQNPJ1T+19/auv2uPknQ4m7WJ3iyNevP2WTgJ2mE8v7Xc2onm85Pz4FbAsED16MTTaF4EySZABAB5dyNY/SJYPbw+ej671T5Nw+5sYz7bnM9vnFxdW7+9Tv+C5cqGrKwJ+e8o+Sg4Fm3RE4noiqkYikiMxATqsQhFBuU7sSHWxRjanokZtKVQG9DnkZiLVcBOoVcEPUJoPYfffXj6TraO4BlpZoTuApcYflJABuIaYBLol0IduQX0+ZQoY2sZ7RnRRNnewN+OpDWE1ok4g1YXTvX0xeFYJuJU/IHGMIAxjakFR9eVVKakFZQ8MEY1AQpjaMN6Dz5Pod4lpNJzQJiMxo66Denzv1FPbMXnruw7FX8nKa9BCURTjj7JKYTigugH9Dan8BnLEwPnPlCI5Bix9op0PaTRj6D/DNofQJlTTemkA2VGrfNK5A4UG3LHidyHYkPuO5ENKDZkw4k8gmJDHkkkYlPSuR3fhGLDN52cH0KxIR86kY+g2JCPnMjHUGzIx07kt1BsyG+dyLtQbMi7TuR9KDbkfSeyBcWGbDmRx1BsyGMncg+KDbknkeUrNYWSEJ2BY1VuQb3IAy1FDC1bTvm2yTrasNsea7pbgnWv6l34a8fueug0KsHuecy70xKse+btg420Y9226B7tJjbsPSf2AGaAHXvgxH4lXpRgv/JYaeclWPdaa0A/O9Ztfb+GJzv2ayf2AdTsWPcedQgtduyhx44xLsEeObEPxcsSrI/VT0uwbrvfBLtix7r3qRb0t2N9rOm0BOu2p4/Bg7Fj3bvVE2i1Y584sU/F6xLsUyf2G7Duduw3Hjvs2xKs2mNXaQfpkz8SwYqtohbmqxJrY6AWOvjH+d4Sk2/cgXYXpp9j+oQZOhH7OWLfE9HIEQ1vubLcjmbk77q5NHNE0xPRyfcmrE2c/Xt5f6zFHojdHLG7gKjySPFdq7FckHehWlzISb5zYc1nTEluv7EWyflQbXkV4rCA4Ll9RjP/JkVLGEGhpqqoneV7PCMDeq5CvKLoTY1S8XDjJrlVMFGvnaiOBdVxot5YUG+cqKkFNXWiLiyoCydKr3wT1/aYAVr/+C5m9MQzgH3k8hKAV7AFu849WKMBzJ8j8AIfUcsh/G1S7O0qVZJhNI/7JGY5nhUscQq1mViDdh0V7lJ8HdMKi0Ay7nkoY3x8wtzGTK45tsLzfCcP8oyJP50BydPP6aC3GNB6qkfnPrXMybvjWj38vXzdq1o9/B5pfE5ePNfq4SdS+sklZG9JbOsS2CasprHUvq7XpcH5F6ah6qu066LFxbc6lHMG6b2uSf9AvpmDS7yXHaqxfnS9Ho3MGF9WGF8dGlrPmaHnelTQe2KvV9WC2iMZybhX1+vKkNAuOpJy6Ke6bwb79OSbUfV6NI7A49qhmHtm1OvO3nE+Gl2vR+Ox4LznnDx5Va9Ho0/PrA9dr0cDsy2hjPN1va5lRw1w7Kzrda36iLLAmAPiOc8t2itKyU+aSmoD8g+qszWmz7+8j2HO5nkeI1RT0r5tOZ1OvpdVS6T8hQis2qSmHOhfTA0frEhjJjad8RXLMCns78t09B6Pmm+AFgNY/XwG4MqZxyChykmg9Y6B4oYz6iqOTOE2nTicJacLqLZsnTi9Rc2Xs0bFthNqdcVlerRaj22y1xnNvTH5hA3SrEsPjdI3XEbRpaFGQUNuenV091au16L215248QJinM+0Lp0I8UladZxq03rT0PE1ecozgcJnPnr+Yrb5VFobjHkSskUoSxVPs5/KI5ltuK/eFDrHzZ8F9EbRXl2Q1RjQiVTmjEJVtpi98Rk9a9rHdCaHPJhGF95jIKmMBZ+aYRYd8+kBWVTT3rp4o75Uho7rGVldZY+r0X0D3beg68c4O7BjPIBaC2KGY3hqeUQ5q7muEtJ4Km7lp6MJvcHqiD4uWEhFg+1NVLCQVVH2WYHKK0DjbOAo3Z/GIh2Fby9Rckf9Nnl07Fq0/Nfo5Fadb4c0x8tnc3kmpkdcN4lrQKuGT3X5aZEDSzCzfrJJ/mv1KJFfHY5oQ11cnxucWS8jOvGPKIIdk2cc02pzrY5ibzM/tfiJ4nQk1Nk5nmYnZCEDsn8B7E8JzcmAfsy7A+oEnS1CTDbSx+4Mcu/G5usMnHNM+3EDwbca9HyLyJZNib+ia66ujOYiRwy8D8wX5rbSSYN8wYi4ptK667VdvfsgUt+TMGcJU9Rz5Trxv0G/1Y+aJ2tLMwI1jG8gk7bO9j4SillQRyHt8tU2SPU1pfw0l+G5lFrvf1qmTwuS7VLEhfLgbt0Dzl16Zl44S1KSO1vqw/toVTYXKY8X9IijPaUonu1+X+7AKPdN2iXXaM21aZb0YRZM8ihC9XVlkRf5VvMqUvejnf1fqGtdF7WGFAOhM7isIVd+P6JozZQyhlnN8/ecVpNd6+lCr2o+I5qLQ2Mtv4PWX8FvJbd69qPTKViFbZoDTEE/aY1wS7DUw4/XdoGXmpmKln7W/PScVL3MlsvE12zddIx9UZvKEc2a1zJroeqXofHCoPHCU4ctOmvUWlTtyhKdOGOLljyt9OVXh1urBuWpk7LbI1OogYeUZizlR7XnpOqO8RXqrZPWupNWCKvVPA0w17wP0r7WF1f3u3x3D8Rd8m265IFx/NKjVTogn0u1VkdqTAE535H21Vz9bWpB7h2yoEiZ73HiiuFTpy6VeS7pb+TOlpCd1xZB3Vt6JfsoG9um+u+WkENaExmtS4W4Qz0iKb8pR7BgkW4bPkdAmf+QfCr2O6pjZrO3fidBwZ/Q8SavKs2LI4UR6d+VeTtYil4PjPg1oJhwKr3rDtCq/4aRAmNUJsHuWWb0hnCX45ME9mg7ZD+X7RSf4o0MiW6T1DPxhYeN4ahXz3VzbqkRq7F9Bj1R6/qt23q4+cXeHF38LnOiF9KuNpQ+6mzh+XK0QrnLFZ+r9DBd4Kv1MaU+ZmSho7wipi0+9+bCEtXjwhgfLvVGUUf+epLXkZlPp3wpq96KcjHTwDbmjOIl1z1QRNi8u+tWb+6GYxydJXodwprUuMVFCbNxicwPmJYWs1JXlvYhbr1SuRvFxk5UtlMo6uZuoe03W8iIrF8sXDkb7m3K3i5EKe4sDFPoCr7RWxYfmjQ/h4K/A2GLDhVHn9xhE/zbLbEj9t7DbYiXss4ZzYBa0Bb0FmLvUI6z2KNaRy8N6iZ9Hw7+PAaga5f0A9pJ68rOlN2Sm9T96b8iK5CKyCm97ll/DCYX90iWOdUZz4Asm3s0A6G+i1N3LIqDz0iKXPz58LmGaxSnQn2nqd4YFHX3CIoc6vBQ9xj83rnuXZ+XyalaX8tcfHnwLqBOXBQOT/7KYxXdz8dCpcYbef8c0DqcVlBXu8X/Og7FR3Oqz8uXW0bfNXvh8da5XyQzsugP118zmpvPbC7n6M8zyUenvSU7P/b7glpvKjFG8/7poz+q54DiNROcB3VLx3hzFml5fanguYBNhkT8U/x5xf1thJc5jTI56lBS5xTl1FQPNzX1jUvb6NRnPjJpOmUyFanpOKJJN2J3xIG4Cz87uQdY93Yof5eS/yLW/v3ZHrSekvVQWXTOHLSpLaLshz5F69Gzvj9bJjHe5eW7vS1owbPwBrXiPd8H1B/v+rYKYyv/Bgmv9a9FInqFiGTxdE+vqw6MoHjyxjkg9T3fgO7ScxaLb54NPc4W1f2pRYlm9In7ZkGnFN8xpOzSXB3Ls3o8OcAb9mGeHwrEb6ktlHYe91wX56NSzkcLnDPSTpHDa+Oz6rtZZVx2DC69PHd2IfslFGfr87zq3OhuKRe+g16N71fg+4aUTdL+OUXCqajO5k0raE6lTOYJ60ioTCTrAePMMH/f1ZHtRQWvC4/x3y9F3zck3QdZOpT/DuiELSV6sdTNHknPNx2rM6n3KqSV36M8ubq2sfh/GSxXjjdv//H2xsM7a19uy//m4EPxifi1uA5L/PfiSyB2JI6BwT9WPl75ZOWXW3/Z+tfWv7f+w10/WJGYX4jCv+2V/wLgpdNy</latexit> E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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Richard Dudley
  64. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit sha1_base64="iqmR2w6gptZhqD4tCYOzKsFQvt8=">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</latexit> Theorem: E(|||ˆ ↵ ˆ||k ||↵ ||k |) = O(n 1 2 ) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">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</latexit> E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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Richard Dudley
  65. Leveraging smoothness in high dimension? <latexit sha1_base64="ePzgW+JqdbDqdwOItscJ4SbXEx4=">AABB1XictVzdchS5FRabv4X8sdnL3HRiSEGKJcZLZbO1lao1tjFeBmyYsWF3DdT8tIeBnulhesbYzPoulds8Qm6Tp8hz5A2Sq7xCzo/UUs+o+6gdgsq2WqPvnKPT0tE5Rxo642SQTVdX/3nhg+99/wc//NGHFy/9+Cc//dnPL3/0i4MsnU268X43TdLJ0047i5PBKN6fDqZJ/HQ8idvDThI/6bzewM+fHMeTbJCOWtPTcfxs2O6PBkeDbnsKTS8uf7yRDsdJfDKYnkZXdq+Nnq9dv/Li8srqzVX6Fy1XbunKitL/9tKPon11qHoqVV01U0MVq5GaQj1RbZVB+VbdUqtqDG3P1BzaJlAb0OexOlOXADuDXjH0aEPra/jdh6dvdesInpFmRugucEngZwLISF0FTAr9JlBHbhF9PiPK2FpGe040UbZT+NvRtIbQOlUvoVXCmZ6hOBzLVB2pP9AYBjCmMbXg6Lqayoy0gpJHzqimQGEMbVjvwecTqHcJafQcESajsaNu2/T5v6gntuJzV/edqX+TlFehRKqpR5/mFNrqmOhH9DZn8BnLkwDnPlCI9Rix9pZ0PaTRj6D/HNofQjmjmtFJB8qcWs8qkRtQfMgNEbkNxYfcFpENKD5kQ0TuQfEh9zQSsRPSuR/fhOLDN0XOj6D4kI9E5GMoPuRjEXkAxYc8EJHfQPEhvxGRd6H4kHdF5H0oPuR9EdmC4kO2ROQ+FB9yX0RuQfEhtzSyfKVOoKREZyCsynWoF3mgpUigZV2U7w5ZRx/2TsCa7pZg5VW9CX/92M0AncYl2K2AeXdUgpVn3jbYSD9WtkX3aDfxYe+J2B2YAX7sjoj9Sr0qwX4VsNJel2DltdaAfn6sbH0fwJMf+0DEPoSaHyvvUbvQ4sfuBuwY4xLsnoh9pN6UYEOs/qQEK9v9JtgVP1bep1rQ348NsaazEqxsTw/Ag/Fj5d3qCbT6sU9E7FN1UoJ9KmK/Buvux34dsMO+K8GaPfYS7SB98kdiWLFV1Nr5qsTaGKi1Bf5Jvrck5Bt3oF3C9HNMnzBDEbGdI7YDEY0c0QiWK8vtaEb+rsylmSOagYhOvjdhbSr27+X9sZYEIDZzxOYCosojxXdtxnJM3oVpkZDTfOfCWsiY0tx+Yy3W86Ha8hrEbgHBc/slzfwbFC1hBIWaqqL2Mt/jGRnRcxXiLUVvZpSGh4yb5lbBRZ2IqI4H1RFRpx7UqYiaeVAzEXXsQR2LKLvyXdxhwAyw+sd3MacnngHsI5eXCLyCddh17sEajWD+7IEX+JhaduFvk2JvqVRJhtE87pOY5XhWsMQTqM3VCrTbqHCT4uuEVlgMknHPXR3j4xPmNuZ6zbEVPst38ijPmITTGZA8/ZwOeosRrad6dO5Tyxl5d1yrh7+Xr3tTq4ffIo2fkRfPtXr4qZZ+eg7ZWxrbOge2CatprLVv63VpcP6FaZj6Jdp10eLiWx3qOYP0TmrS39FvZucc72WDaqwfW69HI3PGlxXGV4eG1XPm6LkeFfSe2Os1taj2SEY67rX1ujKktIuOtBz2qe6bwT49/WZMvR6NPfC4Nijmnjv1urN3nI/G1uvROFCc9zwjT97U69Ho0zPrw9br0cBsS1vH+bZe17KjBjh2tvW6Vn1EWWDMAfGc5xbrFU3IT5ppagPyD6qzNa7Pv7yPYc7meR4jVFOyvm05nU6+l1VLZPyFGKzatKYc6F/MHB+sSGOu1sT4imWYFvb3ZTp2j0fNN0CLEax+PgOQcuYJSGhyEmi9E6B4S4y6iiMzuDURh7PkaAF1qFunordo+XLWqNj2glqluMyO1urxkOx1RnNvTD5hgzQr6aFR+obLKEoaahQ0JNOro7t3er0Wtb8q4sYLiHE+07p0IsQnadVxqk/rTUfHV/UpzxQKn/nY+YvZ5iNtbTDmSckWoSxVPN1+Jo/ktuG+ekPZHDd/FtEbRXt1TFZjQCdSmRiFmmwxe+Nzera09+lMDnkwjS68x0hTGSs+NcMsOubTI7Korr2VeKO+TIaO6xlZXWOPq9F9B933oOvHOBuwYzyEWgtihn14agVEOZdyXaWk8Yn6JD8dTekNVkf0ScFCGhpsb+KChayKsl8WqLwFNM4GjtLDaSzSMfjDJUpy1O+Tx8auRct/lU5uzfl2m+Z4+Wwuz8T0iOsacY1o1fCpLj8tcmAJ5t5P1sh/rR4l8qvDEW2oxPW5w5n1MqIT/5gi2DF5xgmtNml1FHu7+anFTwynPWXOzvE0OyULGZH9i2B/SmlORvTj3h0wJ+hsERKykSF2Z5B7Nz5fZyDOMevHDRTfarDzLSZbNiP+hq67ujKaixwx8D5wtjC3jU4a5AvGxHWirbtd29W7DyLtPQl3ljBFO1euEf/r9Nv8mHmysjQjUMP4BjJt63zvI6WYBXXUpl2+2gaZvq6UV3IZnmup7f5nZbpSkGyTIi6UB3frHnDu0jPzwlkyIbmzpT68j1Zlc5HyeEGPONojiuLZ7vf1Doxy36BdcoXW3CHNkj7MgmkeRZi+UhZ5kW81ryL1MNrZ/4W61XVRa0gxUjaDyxqS8vsxRWuulAnMap6/r2k1+bU+WehVzWdEc3HorOXvoPVX8NvIbZ7D6HQKVuEOzQGmYJ+sRrglWuoRxutOgZeZmYaWfbb87Jw0vdyW88TXbN1sjH1cm8oezZoTnbUw9fPQeOXQeBWowxadNVotmnZjiV6IsUVLn1aG8qvDrVWD8kykLHtkBjUIkNKNpcKo9kSqcoxvUO9EWqsirTasVvc0wF3zIUj/Wl9c3d/lu3uk7pJv0yUPjOOXHq3SAflcprU6UmMKyPm2tq/u6j+kFuTeIQuKlPkeJ64YPnXqUjnLJf2N3tlSsvPWIph7S291H2NjD6n+6RJySGsio3VpELepR6zld+WIFizSTcfniCjz3yafiv2O6pjZ7W3fSVTwJ2y8yavK8uJIYUT6lzJvO0vR644Tv0YUE860d90BWvXfMFJgjMkk+D3LjN4Q7nJ8ksAebYfs57Kd4lO8kSPRTZJ6rv4YYGM46rVz3Z1bZsRmbL+Fnqh1+9Z9PWR+STBHid95TvTatKsNtY86X3g+H6223uWKz1V6mC3wtfqYUR83srBRXhFzqL4I5sIS1ePCmBAu9UZRR/56kteRmU+nQimb3oZyMdPANuYlxUvSPVBE+Ly7a15v7rowjs4SvQ5hXWrcIlHCbFyq8wOupcWs1MWlfYhbL1buRomzE5XtFIa6u1tY+80WMibrlygpZ8O9XdkPC1GKnIVhCl3FN3rL4kOX5hdQ8HekfNGh4RiSO2yCf7uuNtTWe7gN8UbXOaMZUQvagt5C7N3W4yz2qNbRG4e6Sz+EQziPAehakn5AO2ld2ZmyLLlLPZz+W7ICExWL0tue9cfgcpFHssypzngGZNnk0QyU+S5O3bEYDiEjKXIJ58PnGtIojpT5TlO9MRjq8giKHOrwMPcYwt657V2fl8upWl/LXEJ58C5gTlwMDk/+ymMV2y/EQk2cN/L+OaB1OKqgbnaL/3Ucho/lVJ9XKLeMvmv2KuCtc79YZ2TRH66/Ziy3kNlczjGcZ5qPznpLfn7s90W13lTqjOb900d/1M4Bw2uuOA8qS8d4dxZZeUOp4LmAT4ZU/Uf944L8bYQ3OY0yOepQMucU5dRMD5ma+calb3TmsxCZLJ0ymYrUbBzRpBuxG2pH3YWfjdwDrHs7lL9LyX8R6//+bA9aj8h6mCw6Zw4OqS2m7Ic9RevRs70/WyYx3uXlu70taMGz8Aa14j3fh9Qf7/q2CmMr/wYJr/UHKlW9QkSyeLpn11UHRlA8eeMckPmeb0R36TmLxTfPhgFni+b+1KJEc/pEvlnQKcV3HCm7NFfH+qweTw7whn07zw9F6nfU1tZ2HvdcifNeKee9Bc4ZaafI4cT5rPpuVhmXDYdLL8+dHet+KcXZ9jyvOje6WcqF76BX4/sV+L4jZZO0/5oi4YmqzubNKmjOtEzuCetImUwk6wHjzHb+vqsj2+MKXscB479fir7vSLoNsnQo/x3RCduE6CVaN1skPd90rM6k3quQVn+Pkv53g8/pX8SVz27ryue38v/d4GDt5q3f3/z00drKl3f0/3Pwofql+rW6Bmv8M/UlUNtT+8DhVP1V/U39ff3J+tn6n9b/zF0/uKAxH6vCv/W//BeD66fT</latexit> Complexity O(n2)

  66. Leveraging smoothness in high dimension? <latexit sha1_base64="ePzgW+JqdbDqdwOItscJ4SbXEx4=">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</latexit> Complexity O(n2)

  67. Leveraging smoothness in high dimension? <latexit sha1_base64="ePzgW+JqdbDqdwOItscJ4SbXEx4=">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</latexit> Complexity O(n2) <latexit

    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[Vacher, Muzellec, Rudi, Bach, Vialard, 2021] <latexit 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Recent breakthrough:
  68. Overview • Entropic Regularization and Sinkhorn • Convergence Analysis •

    Sinkhorn Divergences • Generative Model Fitting
  69. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit sha1_base64="NVr4d97mahJgu8Mr3K/nifIrenM=">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</latexit> <latexit sha1_base64="NVr4d97mahJgu8Mr3K/nifIrenM=">AAA/zHiczVvNchy3EYadH1vMj+XkmEMmoRRLLkVFMq5yEpeLlkiKpLUSKZEryfZa
  70. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Density fitting: Maximum likelihood (MLE) d↵✓(x) = ⇢✓(x)dx <latexit 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  71. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Density fitting: Maximum likelihood (MLE) d↵✓(x) = ⇢✓(x)dx <latexit 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g✓ X Z ⇣ Generative model fit: ! MLE undefined. ! Need a weaker metric. ↵✓ = g✓,]⇣ <latexit sha1_base64="W/xAH4ZEbKVL5sYCOuVPDm3rS+Q=">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</latexit> <latexit 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<latexit 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<latexit 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↵✓ <latexit 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min ✓ Wp ",p (↵✓, ) <latexit 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  72. Deep Discriminative vs Generative Models ✓1 ✓2 ⇢ ⇢ z

    x g✓ ⇢ ⇢ z x Discriminative Generative d⇠ ⇠1 ⇠2 g✓(z) = ⇢(✓K(. . . ⇢(✓2(⇢(✓1(z) . . .) Deep networks: d⇠(x) = ⇢(⇠K(. . . ⇢(⇠2(⇢(⇠1(x) . . .)
  73. Deep Discriminative vs Generative Models ✓1 ✓2 ⇢ ⇢ z

    x g✓ ⇢ ⇢ z x Discriminative Generative z1 z2 g✓ Z x z X d⇠ d⇠ ⇠1 ⇠2 g✓(z) = ⇢(✓K(. . . ⇢(✓2(⇢(✓1(z) . . .) Deep networks: d⇠(x) = ⇢(⇠K(. . . ⇢(⇠2(⇢(⇠1(x) . . .)
  74. Training Architecture g✓ Z X y1 <latexit 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<latexit 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Stochastic gradient descent <latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="ZvDledmv+Gwz+FxUnlfvrUdxKFc=">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</latexit> min ✓ E(✓) def. = Wp ",p (↵✓, ) <latexit 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<latexit 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✓ ✓ ⌧r ˆ E(✓) <latexit 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<latexit sha1_base64="IBy1TwC0khpcsDQ0WUEsOM5LJ+0=">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</latexit> ˆ E(✓) def. = Wp ",p ( 1 m P i g✓(zi) , ) <latexit 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<latexit 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  75. Training Architecture g✓ Z X y1 <latexit 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<latexit 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Stochastic gradient descent <latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="ZvDledmv+Gwz+FxUnlfvrUdxKFc=">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</latexit> min ✓ E(✓) def. = Wp ",p (↵✓, ) <latexit 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<latexit 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✓ ✓ ⌧r ˆ E(✓) <latexit 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C K . . . ✓1 ✓2 . . . 1m 1/· ⇥mK> ⇥nK 1/· . . . . . . e C/" Input data <latexit 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<latexit 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<latexit 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<latexit 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(y1, . . . , yn) <latexit 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sha1_base64="Lz1gCqAeSuy9yYYGVdEU4UDCj/g=">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</latexit> Generative model <latexit 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<latexit 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<latexit 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(x1, . . . , xm) <latexit 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Sinkhorn, ` = 0, . . . , L 1 <latexit 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ˆ E(✓) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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h(C K)b, ai <latexit 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<latexit sha1_base64="JdisBtcz53UgAh2eCi1NPNdGUeA=">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</latexit> ˆ E(✓) def. = Wp ",p ( 1 m P i g✓(zi) , ) <latexit 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<latexit 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  76. Automatic Differentiation Setup: E : Rn ! R computable in

    K operations. Hypothesis: elementary operations (a ⇥ b, log(a), p a . . . ) and their derivatives cost O(1). Question: What is the complexity of computing rE : Rn ! Rn? function y = E(x) a = xi1*x b = rho(a) z = xi2*b y = 1/2*norm(z-u)^2 ⇢ z x ⇠1 ⇠2 1 2 || · ||2 <latexit 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<latexit 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<latexit 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  77. Automatic Differentiation Setup: E : Rn ! R computable in

    K operations. Hypothesis: elementary operations (a ⇥ b, log(a), p a . . . ) and their derivatives cost O(1). Question: What is the complexity of computing rE : Rn ! Rn? function y = E(x) a = xi1*x b = rho(a) z = xi2*b y = 1/2*norm(z-u)^2 ⇢ z x ⇠1 ⇠2 1 2 || · ||2 <latexit 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<latexit 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<latexit 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Finite di↵erences: K(n + 1) operations, intractable for large n. rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " n) E(✓)) <latexit 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<latexit 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<latexit 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<latexit 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  78. Automatic Differentiation Setup: E : Rn ! R computable in

    K operations. Hypothesis: elementary operations (a ⇥ b, log(a), p a . . . ) and their derivatives cost O(1). Question: What is the complexity of computing rE : Rn ! Rn? function y = E(x) a = xi1*x b = rho(a) z = xi2*b y = 1/2*norm(z-u)^2 function dx = nablaE(x) dz = z-u db = xi2’*dz da = diag(dphi(a)) * db dx = xi1’*da This algorithm is reverse mode automatic di↵erentiation [Seppo Linnainmaa, 1970] Seppo Linnainmaa Theorem: there is an algorithm to compute rE <latexit 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<latexit 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in O(K) operations. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Finite di↵erences: K(n + 1) operations, intractable for large n. rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " n) E(✓)) <latexit 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<latexit 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<latexit 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<latexit 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  79. Examples of Images Generation (a) WGAN MNIST (b) WGAN CelebA

    (c) WGAN L (d) MMD GAN MNIST (e) MMD GAN CelebA (f) MMD GAN Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and datasets. ⇣ Inputs <latexit 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<latexit 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Generated ↵✓ <latexit 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  80. Examples of Images Generation (a) WGAN MNIST (b) WGAN CelebA

    (c) WGAN L (d) MMD GAN MNIST (e) MMD GAN CelebA (f) MMD GAN Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and datasets. ⇣ (a) WGAN MNIST (b) WGAN CelebA (c) WGAN LSUN (d) MMD GAN MNIST (e) MMD GAN CelebA (f) MMD GAN LSUN Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and LSUN bedroom datasets. Inputs <latexit 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Generated ↵✓ <latexit 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<latexit 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h⇠ <latexit 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<latexit 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<latexit 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! Performance evaluation of generative models is an open problem. ! Influence of "? <latexit 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<latexit 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<latexit 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! Need to learn the metric d(x, y) = ||h⇠(x) h⇠(y)|| (GANs) <latexit 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<latexit 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<latexit 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Ian Goodfellow " <latexit 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<latexit 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Inception score <latexit 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  81. Progressive Growing of GANs for Improved Quality, Stability, and Variation

    Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018 X “fake” <latexit 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<latexit 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g✓
  82. Progressive Growing of GANs for Improved Quality, Stability, and Variation

    Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018 X “fake” <latexit 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<latexit 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g✓
  83. Open Problems Monge Kantorovic Dantzig Brenier Villani Figalli Toward high-dimensional

    OT: <latexit 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<latexit 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<latexit 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<latexit 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! Scalable geometrical loss functions in high dimension? <latexit 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<latexit 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<latexit 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! Performance quality measures for unsupervised learning? <latexit 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<latexit 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<latexit 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  84. Open Problems Monge Kantorovic Dantzig Brenier Villani Figalli Toward high-dimensional

    OT: <latexit 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<latexit 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<latexit 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<latexit 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! Scalable geometrical loss functions in high dimension? <latexit 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<latexit 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<latexit 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! Performance quality measures for unsupervised learning? <latexit 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<latexit 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<latexit 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<latexit 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Metric learning for OT: <latexit 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<latexit 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<latexit 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z g✓ z x d⇠ ! Adversarial training to <latexit 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<latexit 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<latexit 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<latexit 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leverage multi scale priors? <latexit 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<latexit 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  85. Open Problems Monge Kantorovic Dantzig Brenier Villani Figalli Toward high-dimensional

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<latexit 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<latexit 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<latexit 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! Scalable geometrical loss functions in high dimension? <latexit 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<latexit 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<latexit 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! Performance quality measures for unsupervised learning? <latexit 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<latexit 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<latexit 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<latexit 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Metric learning for OT: <latexit 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<latexit 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<latexit 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z g✓ z x d⇠ ! Adversarial training to <latexit 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<latexit 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<latexit 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<latexit 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leverage multi scale priors? <latexit 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<latexit 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Beyond comparing measures: <latexit 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! Using Gromov-Wasserstein geometry? <latexit 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<latexit 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<latexit 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! Learning for surfaces, graphs, metric spaces? <latexit 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<latexit 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<latexit 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