A Mathematical Journey of Regularization on Measures

In this presentation, we will take a tour of the path of regularization on measures. We will use an example to revisit recent developments in this tool at the intersection of statistics, learning, and optimization. Along the way, we will encounter various species such as the dual of a convex program with its subgradient, empirical process concentration with its "golfing scheme," kernel functions with their Hilbertian structures, stochastic gradient descent with particles, and beautiful weather days with an optimal transport distance called partial displacement. The landscapes traversed will lead us to discuss applications in unsupervised learning (deep or not), super-resolution, tensor processing or even quadrature. We will avoid technical details on sensitive topics, but any questions about these aspects are welcome. No technical equipment is required; a regular practice of indulgence is sufficient.

Transcript

1. Yohann DE CASTRO (Institut Camille Jordan, Centrale Lyon) A Journey of Regularization on Measures November 2023 DATA-GAIA seminar, Grenoble 1 Based on joint works with C. Marteau (ICJ), S. Gadat (TSE), J.M. Azaïs & F. Gamboa (Toulouse 3), C. Maugis (INSA Toulouse), D. Henrion & J.B. Lasserre (LAAS), R. Gribonval (INRIA, ENSL), N. Jouvin (INRAE), C. Boyer (P6), J. Salmon (Univ. Montpellier)
2. DATA-GAIA 23 Y. De Castro (ICJ) Featuring some Functional spaces 2 <latexit sha1_base64="K3t+fy9ExkYBnZLm7VAXeLvKjH4=">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</latexit> • Observation: Y 2 H with H separable Hilbert space;
3. DATA-GAIA 23 Y. De Castro (ICJ) Featuring some Functional spaces 2 <latexit sha1_base64="K3t+fy9ExkYBnZLm7VAXeLvKjH4=">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</latexit> • Observation: Y 2 H with H separable Hilbert space; <latexit sha1_base64="L1d41cDhegDcpLbeWNh3YN4xP78=">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</latexit> • Measures: (X, dX ) compact metric space, (M(X), k · k1) space of measures with bounded TV norm; <latexit sha1_base64="gD4B+SlOk3nReFi6IVpvQwzhIYs=">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</latexit> (M(X), k · k1) = (C(X), k · k1)? <latexit sha1_base64="sOLTws5yYhzmG4sMq19M3Siv+OU=">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</latexit> hµ, fiM(X),C(X) = Z X fdµ
4. DATA-GAIA 23 Y. De Castro (ICJ) Featuring some Functional spaces 2 <latexit sha1_base64="K3t+fy9ExkYBnZLm7VAXeLvKjH4=">AAAlDHicpZpbbxtFFMen5VbKLYVHXiziSq2EqiQSFxWEStNLeo3bpknaOkQ+9treZn3p7tpJauUr8Gl4Q7whXvgEfAPgFT4AZ87M2uuZv70uTZRkfPY3Z247/51zNtSPwiRdWfnj1Ok33nzr7XfOvHv2vfc/+PCjpXMfbye9QVwPHtd7US/epVoSRGE3eJyGaRTs9uOg1qEo2KGDdX19ZxjESdjrbqXH/WCvU2t1w2ZYr6Vs2l/6oVylQRQFabm0SUkQD8V+uVSudmppm5qlJ9Wwa8pU2iiXDsO0nV0Uw6jKUBL0a3GN2zwpbYQRBXFaSvq1evDN/tLyyqUV+Sr5hVVbWFb2q9I7d+53VVUN1VN1NVAdFaiuSrkcqZpK+PuZWlUrqs+2PTViW8ylUK4H6kSd5boDpgImamw94N8t/vTMWrv8WftMpHadW4n4J+aaJXWe6/SYi7msWyvJ9YF41tZZvkfiU/ftmP+S9dVha6rabC2ql5GL1tNjSVVTfS1jCHlMfbHo0dWtl4HMiu55KTeqlD302abLDb4ec7kuNbN5LkmdRMau57Ym1/8UUlv157plB+qvub1M1JBL0zN+NDXns2emwdYmU7qdeVxXZot4LGQ5U474d4+ZWEbc5pVJmDuW0Y3EVhPmYHzXdLl0KDMfSO/0nIyczyds2XIs52WkZh5D6fEe+9Pf5+0sRnZWU5np2M5Zg8tN/ltX37PX6njdTY0RW09szybkVUheBeQ6JNcBeQ2S1wB5HZLXAXkDkjcAeROSNwG5AckNQN6C5C1A3obkbUDegeQdQN6F5F1A3oPkPUDeh+R9QG5CchOQFUhWAPkAkg8A+RCSDwH5CJKPALkFyS1APobkY0BuQ3IbkDuQ3AHkLiR3AfkEkk8A+RSSTy1pFIZETxuexpTUBdbM/ljv6/L8DNTFqTbIqk/mJXuG9WZoEFkNwryvRGSVCPO+HpHVI8z7qkRWlTDvaxNZbcK8r1BkFQrzvk6R1SnM+2pFVq0w72sWWc3CvK9cZJUL875+kdUvzPsqRlbFMO9rGVktw7yvaGQVDfO+rpHVNcz76kZW3TDvaxxZjcO8r3RklQ7zvt6R1TvM+6pHVvUw72sfWe3DvK+AZBUQ874OktVBzPtqSFYNMe9rIllNxPxEGYu1UZ9BDwu0sTalv8RWfXb0+0SAI8DVAVcHXANwDcAFgAtg/5qAbAKyBbgW4NqAawMuBFwIuOeAew64A8AdAC4CXAS4DuA6gOsCrgu4HuB6gOsDrg+4F4B7AbgYcDHgEsAlgEsBlwJuALgB4IaAGwLuEHCHgDsC3BHgjgF3DLiXgHs545zVktgv4PsOxXUmjtRRbkf8de1uJjm96VhUK8hopopVp7iTAp8k7Rd5nFBF/loSrXcKPea5Ip8xj6VX4C9jinxF0q5WyKIeTpNFfjty187z18nd17P9HIm6zfNzlNO/2X5M3qF4ZfPcbJ/mHBLyqnXkmTbb34Qq8rcpWZ1Wgb8JVeRvazySef62Fh5vstB4k4XHG4hmz9tj5jRUErV4LnUz5aC5GSB9fToH1BCdRtEXOTkgQ6K4i5wckCFRxEVODsiQKNYiJwdkSBRlkZMDMiSKr8jJARkSRVbk5IAMiWIqcnJAhkTRFDk5IEOiOIqcHJAhUQRFTg7IkCh2IicHZEgUNZGTAzIkipfIyQEZEkVK5OSADIliJHJyQIZE0RE5OSBDoriInByQIVFERE4OyJAoFiI1iYimaWNHNbYBjeIgcnJBhkQREDm5IEOi2IecXJAhUdRDTi7IkNOZoD4rS0/OGSS5/1Ay3vOeM9fl6VJlu8kSad86s53l7GftddM7fQqa5bnyyp4rC3rWJ6PQeh8q/Y4isxTVJKGzfg2lX5G8PwkWqN2W3prah1xLr01mK6pr3sU0pJ18/bx93jptT/V81oxm1Hxf6zL2Il8ZVeyruF8ZNd9XwCcj7aEuM6rf8u3NuP+q8ixtsvWCWmZOz71+irfZcrGglb70ZJb3ymt6H9oVmPgvXq3Xaa9uV2mR9rIVfd32Fh1ftur/t72q7MyGPXdVrbql9g6ZrJx5k5edrRA58RPYUleeHoYNREFcxcV1dP+X1drUnaP3vck2xBJz6N6Wx2RZZqAj7xn17KxJHDEaX880b7rt0M53Iidd/b4z30qHtd4wJ5YKlZ+zuCpjMOqj91Yq58QQZC3Ms7HL15oOn7/i1oqkd72CfhrG9FOvkp+7aL2yn1bOj3ny9WwcFAs5OZXPj8i/HfdL21pq8h8E8+7/rvTlxJmpMt/d5XFv9Xv4Mt/d5YWeaN9Zb/HC/dCr2JXx6r24ZvdifhwlucM+l79r/PtVvBsu7znv193PWRtoX2drrPdRdgrb8XZah6/q53BgGd3Gw7EvrZKx5GpLcs2t3bR3rb561/YsLqh9TbQhkqsBn7fNTt50tCzbWy3ZM6NcOd+DitTRecS8KjTFj84QPQI7fBJ19mQVzMxMrPuOwmjvq7nzke8nhj78DKB+S5rIiSmQsY+sDbVouLFC7S8tr7r/t+MXttcurX556YsHa8tXLtv/6TmjPlWf8d2yqr5SVzheq3B7dfWr+lv9o/4t/1j+qfxz+ReDnj5l63yipr7Kv/0HNqtSvQ==</latexit> • Observation: Y 2 H with H separable Hilbert space; <latexit 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• Measures: (X, dX ) compact metric space, (M(X), k · k1) space of measures with bounded TV norm; <latexit sha1_base64="gD4B+SlOk3nReFi6IVpvQwzhIYs=">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</latexit> (M(X), k · k1) = (C(X), k · k1)? <latexit sha1_base64="sOLTws5yYhzmG4sMq19M3Siv+OU=">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</latexit> hµ, fiM(X),C(X) = Z X fdµ <latexit sha1_base64="C0HShmYcdwoIqhd/3Mhlaxcxl/8=">AAAlE3icpZrJbhtHEIbbWR1nk5NjLkREAzZgGJKALDCCwLG8b5JtWZJtCgKLHFJjDWfomaEWE3qMPE1uQW5GHiBvkBxzTS6pru4hh90/OXQsQVKz++vq/Z+uGlE/CrN8aemPU++8+977H3x4+qMzH3/y6WefL5z9YjNLBmkreNJKoiTdpmYWRGEcPMnDPAq2+2nQ7FEUbNH+qi7fOgjSLEzijfy4H+z0mt047IStZs5ZuwtBI07CuB3E+Zl6gwZRFOT12o2gmQ/SILtcqzfaYdaPmsdZfhwFtcZBM+3vhY2LlxsXjxph3Og1871WM6ptc6qf5YkFdkeFRLVb9druwuLSpSX5qvmJZZtYVPZrPTl79rVqqLZKVEsNVE8FKlY5pyPVVBl/P1fLakn1OW9HDTkv5VQo5YE6UWe47oCpgIkm5+7z7y5/em5zY/6sbWZSu8WtRPyTcs2aOsd1EuZSTuvWalI+EMs6d5rtodjUfTvmv2Rt9Tg3V3ucW1WvIOetp8eSq476XsYQ8pj6kqNH17JWBjIruue10qhyttDnPJ1uc3nK6ZbULOa5JnUyGbue26aU/ymkztWfW5YdqL9m9jJTB5yanPGjiTmfPjNtzu0wpduZxcUyW8RjIcuZdMS/E2ZSGfEer0zG3LGMbih5TWH2R7sm5tShzHwgvdNzMnQ+n3DOhpNzTkZq5jGUHu+wPf19zs5iZGc1l5lO7Zy1Od3hvy31E1ttjNbd1Bhy7ont2Zi8CsmrgFyF5Cogr0HyGiCvQ/I6IG9A8gYgb0LyJiBvQfIWIG9D8jYg70DyDiDvQvIuIO9B8h4g70PyPiAfQPIBINcguQbIdUiuA/IhJB8C8hEkHwHyMSQfA3IDkhuAfALJJ4DchOQmILcguQXIbUhuA/IpJJ8C8hkkn1nSKAyJnrY9jamp86yZ/ZHet+T5GagLE22QVZ/CSvEMS6ZoEFkNwryvRGSVCPO+HpHVI8z7qkRWlTDvaxNZbcK8r1BkFQrzvk6R1SnM+2pFVq0w72sWWc3CvK9cZJUL875+kdUvzPsqRlbFMO9rGVktw7yvaGQVDfO+rpHVNcz76kZW3TDvaxxZjcO8r3RklQ7zvt6R1TvM+6pHVvUw72sfWe3DvK+AZBUQ874OktVBzPtqSFYNMe9rIllNxPxYGau1Ud9BDyu0sTmhv8S5+u7o94kAR4BrAa4FuDbg2oALABfA/nUA2QFkF3BdwO0Bbg9wIeBCwL0A3AvA7QNuH3AR4CLA9QDXA1wMuBhwCeASwPUB1wfcS8C9BFwKuBRwGeAywOWAywE3ANwAcAeAOwDcIeAOAXcEuCPAHQPuGHCvAPdqyj2rK75fwPsO+XXGj9Rebk/sxfY0k9zetC+qFWQ4VcUaE9xJhU2S9qssjqkqe13x1nuVFstclc2Ux5JU2CuYKluRtKsVsqqHk2SV3Z7s2ln2eqV9Pd3OkajbLDtHJf2bbsfEHapXtsxNt2nuISGvWk+eadPtjakqe2sS1elW2BtTVfY2RiOZZW9j7vFmc403m3u8gWj2rDNmbkM1UYsXUrdQDpoZAdLlkzGgtug08r7IiQEZEvld5MSADIk8LnJiQIZEvhY5MSBDIi+LnBiQIZF/RU4MyJDIsyInBmRI5FOREwMyJPKmyIkBGRL5UeTEgAyJPChyYkCGRL4TOTEgQyKviZwYkCGRv0RODMiQyFMiJwZkSOQjkRMDMiTyjsiJARkS+UXkxIAMiTwicmJAhkS+EKmxRzRJm3xUYxPQyA8iJxZkSOQBkRMLMiTyfciJBRkSeT3kxIIMORkJ6rOyJHLPIIn9hxLxnvWcuS5PlwbnmyiRtq0j20XMftpZN73Tt6Bpltff2PL6nJb1zSi01g+UfkdR5FTVJKGLfh1IvyJ5fxLMUXtPemtqH3ItvTZFXlVd8y6mLe2U65fzZ63T5kTPp81oQc22tSpjr7JVUNW2qvtVULNtBXwz0hZaMqP6Ld/OlP3XkGdph3PPq0Xm9Nzrp/ge51yoaKUvPZlmff0trR/YFRjbr16tt2mvZVdpnvaKFX3b9uYdX7Hq/7e9hpzMtr13Nay65XaHjFfOvMkr7laIHNsJbCqWp4dhA1EQV3FxHd3/RbUysXP0uTfRhlR8Dt3b+oisywz05D2jnp0V8SOGo/JC8ybbDu18Z3LT1e87y630WOsNc2KpUPkxi6syBqM++mzlck8MQdTCPBtjLus4fLnErRVJ75KKfhrG9FOvkh+76L6xnW7JjnnyJdYPSoUc38pne+Q/jPql87pq/B8Es/Z/LH05cWaqzru7Puqtfg9f591dn+uJ9qO1ls7dD72KsYxXn8UVexbL46jJDrsof1f495tYN1zZctmue56LNtC5LtZYn6PiFrblnbQel+rncGAZ3cajkS2tkqnEamtS5tbu2F2rS+/ZnqUVta+JNkRSGvB925zkNUfLirPVlTMzLKXLPViXOjqOWFaFjtjREaLH4ISPvc5EVsHMzDh311EYbX25dD/y7aTQhh8B1G9JM7kxBTL2oc1DLRpupFC7C4vL7v/t+InNlUvL31765uHK4pXL9n96Tquv1Ne8W5bVd+oK+2vr3F5LvVZ/q3/Uv/Wf67/Uf63/ZtB3Ttk6X6qJr/rv/wErUVdb</latexit> • Features: ' : x 2 X 7! 'x 2 H <latexit sha1_base64="itENhbte+jL/xZD8T9WKnWjBpDY=">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</latexit> Assumption: ' is continuous
5. DATA-GAIA 23 Y. De Castro (ICJ) 3 Measure embedding <latexit sha1_base64="aJDnvLFL3xtlCZZ2ao5TCU5MSAw=">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</latexit> : µ 2 M(X) 7! Z X 'xdµ(x) <latexit sha1_base64="itENhbte+jL/xZD8T9WKnWjBpDY=">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</latexit> Assumption: ' is continuous
6. DATA-GAIA 23 Y. De Castro (ICJ) 3 Measure embedding <latexit sha1_base64="aJDnvLFL3xtlCZZ2ao5TCU5MSAw=">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</latexit> : µ 2 M(X) 7! Z X 'xdµ(x) <latexit sha1_base64="itENhbte+jL/xZD8T9WKnWjBpDY=">AAAk8nicpZrZbttGFIYn6ZamW9Je9kaoFSABisA20AVBUWRx9sVO4i2JgkBHomTGFKmQlGxH8Iv0ruhd0TfpTW/TJ2jfomfODCVq5peoNDZsj2a+ObP/nHNo6kdhli8vvzlx8r33P/jwo1Mfn/7k088+/+LM2S+3s2SQtoKtVhIl6S41syAK42ArD/Mo2O2nQbNHUbBD+9d0+c4wSLMwiTfzo37wvNfsxmEnbDVzznpxZq0RJ2HcDuL8dCMPDnPqjK5k2aDX18WXjmv1RjvM+lHzKMuPoqDWGDbT/l5Yr4VZrZXEeRgPkkH24szS8sVl+ar5iRWbWFL2ayM5e/ZP1VBtlaiWGqieClSsck5Hqqky/n6mVtSy6nPeczXivJRToZQH6lid5roDpgImmpy7z7+7/OmZzY35s7aZSe0WtxLxT8o1a+oc10mYSzmtW6tJ+UAs69xZtkdiU/ftiP+StdXj3FztcW5VvYJctJ4eS6466kcZQ8hj6kuOHl3LWhnIrOie10qjytlCn/N0us3lKadbUrOY55rUyWTsem6bUv6PkDpXf25ZdqD+ndvLTA05NT3jh1NzPntm2pzbYUq3M4+LZbaIx0KWM+mIfyfMpDLiPV6ZjLkjGd1I8prC7I93TcypA5n5QHqn52TkfD7mnE0n55yM1MxjKD1+zvb09zk7i5Gd1VxmOrVz1uZ0h/+21BW22hivu6kx4txj27MJeRWSVwF5DZLXALkGyTVAXofkdUDegOQNQN6E5E1A3oLkLUDehuRtQN6B5B1A3oXkXUDeg+Q9QN6H5H1APoDkA0CuQ3IdkBuQ3ADkQ0g+BOQjSD4C5GNIPgbkJiQ3AbkFyS1AbkNyG5A7kNwB5C4kdwH5BJJPAPkUkk8taRSGRE/bnsbU1HnWzP5Y71vy/AzUhak2yKpPYaV4hiUzNIisBmHeVyKySoR5X4/I6hHmfVUiq0qY97WJrDZh3lcosgqFeV+nyOoU5n21IqtWmPc1i6xmYd5XLrLKhXlfv8jqF+Z9FSOrYpj3tYyslmHeVzSyioZ5X9fI6hrmfXUjq26Y9zWOrMZh3lc6skqHeV/vyOod5n3VI6t6mPe1j6z2Yd5XQLIKiHlfB8nqIOZ9NSSrhpj3NZGsJmJ+oozV2qjvoAcV2tic0l/iXH139PtEgCPAtQDXAlwbcG3ABYALYP86gOwAsgu4LuD2ALcHuBBwIeBeAu4l4PYBtw+4CHAR4HqA6wEuBlwMuARwCeD6gOsD7hXgXgEuBVwKuAxwGeBywOWAGwBuALgh4IaAOwDcAeAOAXcIuCPAHQHuNeBez7hndcX3C3jfIb/O+JHay+2JvdieZpLbm/ZFtYKMZqpYY4o7rrBJ0n6VxQlVZa8r3nqv0mKZq7KZ8liSCnsFU2Urkna1Qlb1cJqsstuTXTvPXq+0r2fbORR1m2fnsKR/s+2YuEP1ypa52TbNPSTkVevJM222vQlVZW9dojrdCnsTqsre5ngk8+xtLjzebKHxZguPNxDNnnfGzG2oJmrxUuoWykFzI0C6fDoG1BadRt4XOTEgQyK/i5wYkCGRx0VODMiQyNciJwZkSORlkRMDMiTyr8iJARkSeVbkxIAMiXwqcmJAhkTeFDkxIEMiP4qcGJAhkQdFTgzIkMh3IicGZEjkNZETAzIk8pfIiQEZEnlK5MSADIl8JHJiQIZE3hE5MSBDIr+InBiQIZFHRE4MyJDIFyI18YimaZOPamwDGvlB5MSCDIk8IHJiQYZEvg85sSBDIq+HnFiQIacjQX1WlkTuGSSx/1Ai3vOeM9fl6dLgfBMl0rZ1ZLuI2c8666Z3+hY0y/LGW1veWNCyvhmF1vpQ6XcURU5VTRK66NdQ+hXJ+5Nggdp70ltT+4Br6bUp8qrqmncxbWmnXL+cP2+dtqd6PmtGC2q+rWsy9ipbBVVtq7pfBTXfVsA3I22hJTOq3/I9n7H/GvIs7XDuebXEnJ57/RTf45wLFa30pSezrG+8o/WhXYGJ/erVepf2WnaVFmmvWNF3bW/R8RWr/n/ba8jJbNt7V8OqW253yGTlzJu84m6FyImdwKZieXoYNhAFcRUX19H9X1KrUztHn3sTbUjF59C9rY/JusxAT94z6tlZFT9iNC4vNG+67dDOdyY3Xf2+s9xKj7XeMMeWCpUfs7gqYzDqo89WLvfEEEQtzLMx5rKOw5dL3FqR9C6p6KdhTD/1Kvmxi+5b2+mW7JgnX2L9oFTIya18vkf+07hfOq+rJv9BMG//x9KXY2em6ry76+Pe6vfwdd7d9YWeaD9ba+nC/dCrGMt49VlctWexPI6a7LBv5e8q/34b64YrWy7bdc9z0QY618Ua63NU3MJ2vJPW41L9HA4so9t4NLalVTKVWG1NytzaHbtrdek927O0ovaaaEMkpQHft81JXne0rDhbXTkzo1K63IMNqaPjiGVV6IgdHSF6DE74xOtMZBXMzExyXzgKo62vlO5Hvp0U2vAjgPotaSY3pkDGPrJ5qEXDjRXqxZmlFff/dvzE9urFle8vfvdwdenyJfs/PafU1+ob3i0r6gd1mf21DW6vpX5Xf6k36u96Xv+l/mv9N4OePGHrfKWmvup//AdQ+Us4</latexit> Assumption: ' is continuous <latexit sha1_base64="dI+1oLveTiCfe40gg3jlkIFnma8=">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</latexit> • Well-defined and bounded linear; <latexit sha1_base64="5uBeJvWNeGWvmYts+24cKf4FS8A=">AAAll3icpZrrbhvHFcdHadOkbtrY7aeiXxYVCdiAakgCekEapKlvcRI7om1ZUpI1hD3kklxrL8zuUpcQah8wT5A3aN6iZ87MksvZP7l0LEHScOY3Z+7/nXNWNImjotzd/XHrnV/88t1fvff+r2/85oPf/u7Dm7d+f1Rk07wfvuxncZafUFCEcZSGL8uojMOTSR4GCcXhMZ3d1+XH52FeRFl6WF5NwldJMEqjYdQPSs46vfmTn2ZROgjT8ka349M0jsOy4x2HcfyXQThkowMvSAceZVOGBp5uJsj/eaNCfc8fZnkQx97Yj1I/CcoxkffY39lZlPjJtCrrB7H39PY8eXKHwY5ueBAVkzi4KsqrOPR748jf+cjfWVsxCSZFmXkMlKezRcG1fx7kk3F0eil5eeIN2Mztyzud05vbu3d35ctrJvZsYlvZr15269YPylcDlam+mqpEhSpVJadjFaiCv79Ve2pXTTjvlZpxXs6pSMpDda1ucN0pUyETAeee8e8Rf/rW5qb8WdsspHafW4n5J+eanupynYy5nNO6NU/Kp2JZ566yPRObum9X/JesrYRzSzXm3LZ6FblpPT2WUg3VP2QMEY9pIjl6dH1rZSqzonvu1UZVsoUJ5+n0gMtzTvelZjXPntQpZOx6bgMp/5+QOld/7lt2qn5a28tCnXNqecYvl+Z89cwMOHfIlG5nHZfKbBGPhSxn0jH/zpjJZcRjXpmCuSsZ3UzyAmHO5rsm5dSFzHwovdNzMnM+X3POoZPTlZGaeYykx6/Ynv7u2lmM7ayWMtO5nbMBp4f8t6/+zVb9+bqbGjPOvbY9W5D3IHkPkPcheR+QDyD5AJAPIfkQkI8g+QiQn0HyM0A+huRjQH4Oyc8B+QUkvwDkl5D8EpBPIPkEkE8h+RSQX0HyK0AeQPIAkD1I9gD5DJLPAPkcks8B+QKSLwB5CMlDQL6E5EtAHkHyCJDHkDwG5AkkTwD5NSS/BuQ3kPzGkkZhSPR00NAYT91mzZzM9b4vz89Q3Vlqg6z6VFaqZ1i2QoPIahDmm0pEVokw39QjsnqE+aYqkVUlzDe1iaw2Yb6pUGQVCvNNnSKrU5hvqhVZtcJ8U7PIahbmm8pFVrkw39QvsvqF+aaKkVUxzDe1jKyWYb6paGQVDfNNXSOra5hvqhtZdcN8U+PIahzmm0pHVukw39Q7snqH+abqkVU9zDe1j6z2Yb6pgGQVEPNNHSSrg5hvqiFZNcR8UxPJaiLmF8rYro36DnrRoo3Bkv4S5+q7Y7NPBDgCXB9wfcANADcAXAi4EPZvCMghIEeAGwFuDLgx4CLARYB7DbjXgDsD3BngYsDFgEsAlwAuBVwKuAxwGeAmgJsA7jvAfQe4HHA54ArAFYArAVcCbgq4KeDOAXcOuAvAXQDuEnCXgLsC3BXgvgfc9yvuWSPx/ULed8ivM36k9nITsZfa00xye9O+qFaQ2UoV85e46xabJO23WVxQbfZG4q0nrRbrXJvNnMeStdirmDZbsbSrFbKth8tkm91Edu06e0ltX6+2cynqts7OZU3/VtsxcYf2la1zq22ae0jEq5bIM221vQXVZu9AojqjFnsLqs3e4Xwk6+wdbjzeYqPxFhuPNxTNXnfGzG3IE7V4LXUr5aC1ESBdvhwDGohOI++LnBiQIZHfRU4MyJDI4yInBmRI5GuREwMyJPKyyIkBGRL5V+TEgAyJPCtyYkCGRD4VOTEgQyJvipwYkCGRH0VODMiQyIMiJwZkSOQ7kRMDMiTymsiJARkS+UvkxIAMiTwlcmJAhkQ+EjkxIEMi74icGJAhkV9ETgzIkMgjIicGZEjkC5FaeETLtMlHNY4AjfwgcmJBhkQeEDmxIEMi34ecWJAhkddDTizIkMuRoAkrSyb3DJLYfyQR73XPmYfydPE530SJtG0d2a5i9qvOuumdvgWtstx7Y8u9DS3rm1FkrZ8r/Y6iymmrSUJX/TqXfsXy/iTcoPZYemtqX3AtvTZVXltd8y5mIO3U69fz163T0VLPV81oRa23dV/G3marotpttferotbbCvlmpC30ZUb1W75XK/afL8/SIefeVtvM6bnXT/Ex59xpaWUiPVllvfeW1s/tCizst6/W27TXt6u0SXvVir5te5uOr1r1n9ueLydzYO9dvlW30u6QxcqZN3nV3QqRCzuhTaXy9DBsKAriKi6uo/u/rfaXdo4+9ybakIvPoXvbmZMdmYFE3jPq2dkXP2I2L680b7ntyM53ITdd/b6z3krCWm+Ya0tFqhmzuCdjMOqjz1Yp98QIRC3MszHlsqHD10vcWrH0Lmvpp2FMP/UqNWMXoze2M6rZMU++zPpBuZCLW/l6j/zjeb903kgt/oNg3f5PpS/Xzkx1eHd35r3V7+E7vLs7Gz3RPrHW8o37oVcxlfHqs7hvz2J9HJ7ssB35u8+/38S64eqW63bd81y1gc51tcb6HFW3sOPGSUu4VD+HQ8voNp7PbWmVzCVW60mZW3tod60ufWJ7lrfUfiDaEEtpyPdtc5IPHC2rztZIzsyslq73oCd1dByxrgpDsaMjRC/ACV94nZmsgpmZRe6pozDa+l7tftS0k0MbzQigfktayI0plLHPbB5q0XBzhTq9ub3n/t9OM3G0f3fvb3f/+mx/+9OP7P/0vK/+pP7Mu2VP/V19yv5aj9vrbx1sTbf+s/Xf7h+7/+o+6j426Dtbts4f1NJX99n/AZy4gVc=</latexit> • 8h 2 H , 8µ 2 M(X) , <latexit sha1_base64="aQyWDgWJ71C3fvl1NWXHw8/QAIM=">AAAmLXicpZpbc9vGFcfX6i11erF7eeoLpiJnrBnFI2mmaSdtOqllJ05ix7QtS0oCR8NDgCQsEGAAUJcw7FfqB+lzHzrT6Vun7bfo2bMLElz8SdC1PBaXi9+ePbuL/WPPgWgcR3mxt/f3G1vf+e73vv+Dt3548+0f/fgnP711+2fHeTrJeuGLXhqn2Sl18zCOkvBFERVxeDrOwu6I4vCEzg/19ZOLMMujNDkqrsfhy1F3kET9qNctuOrs9o0/+0kaJUGYFDfbLZ8mcRwWLe8kjON3grDPVgOvmwQepROGAk/3081+f7NEfc/vDKOv/LzoZv7ue/7uHX/ULYZE3sNd/1u/F6SF/+3ZdF452/GL1Ne2Mm1q6lPfM0163dg7XBRPd5baR0m/uJ7tzFo1T9jrIMrHcfc6L67j0I+7yYA/2Ct/NPF3d/3doZ9JXdWN9z02Wdga6W9WtrzoZuNhdHa1qqm/K+Vs5AXcw52rnffLlra/xYx4Tnvd0ePdxXBnrbNb23t39+THqxf2bWFb2Z9Oevv2X5WvApWqnpqokQpVogoux6qrcv73pdpXe2rMdS/VlOsyLkVyPVQzdZPbTpgKmehy7Tn/HvC3L21twt+1zVxa97iXmP9n3NJTbW6TMpdxWffmyfWJWNa1q2xPxab27Zo/ydoacW2hhlzb1K4kN22nx1KovvqdjCHiMY2lRo+uZ61MZFa0515lVAVbGHOdLgd8PeNyT1qW8+xJm1zGrue2K9f/I6Su1d97lp2o/671MlcXXFqe8aulOV89MwHX9pnS/azjEpkt4rGQ5Uw55t8pM5mMeMgrkzN3LaObSl1XmPP5XZNw6VJmPhTv9JxMne8zrjlyatoyUjOPkXj8ku3pf207i7Gd1UJmOrNzFnC5z5899Se26s/X3bSYcu3MerYg70HyHiAPIXkIyPuQvA/IB5B8AMgPIfkhID+C5EeAfAjJh4D8GJIfA/ITSH4CyE8h+SkgH0HyESAfQ/IxID+D5GeAfALJJ4DsQLIDyKeQfArIZ5B8BsjnkHwOyCNIHgHyBSRfAPIYkseAPIHkCSBPIXkKyM8h+Tkgv4DkF5Y0CkOip0FNYzx1hzVzPNf7njw/Q7Wz1AdZ9SmtlM+wdIUGkdUgzNeViKwSYb6uR2T1CPN1VSKrSpivaxNZbcJ8XaHIKhTm6zpFVqcwX1crsmqF+bpmkdUszNeVi6xyYb6uX2T1C/N1FSOrYpivaxlZLcN8XdHIKhrm67pGVtcwX1c3suqG+brGkdU4zNeVjqzSYb6ud2T1DvN11SOrepivax9Z7cN8XQHJKiDm6zpIVgcxX1dDsmqI+bomktVEzC+UsVkb9Rn0skEbu0v6S1yrz451nwhwBLge4HqACwAXAC4EXAj96wOyD8gB4AaAGwJuCLgIcBHgXgHuFeDOAXcOuBhwMeBGgBsBLgFcArgUcCngxoAbA+5rwH0NuAxwGeBywOWAKwBXAG4CuAngLgB3AbhLwF0C7gpwV4C7Btw14L4B3DcrzlkDif1Cvu9QXGfiSB3ljsReYnczyelNx6JaQaYrVcxf4mYNNkn6b7K4oJrsDSRaHzVarHJNNjMeS9pgr2SabMXSr1bIJg+XySa7I7lr19kbVe7r1XauRN3W2bmq6N9qOybv0LyyVW61TXMOiXjVRvJMW21vQTXZeyJZnUGDvQXVZO9oPpJ19o42Hm++0Xjzjccbimav22PmNOSJWryStqVy0NoMkL6+nAMKRKdR9EVODsiQKO4iJwdkSBRxkZMDMiSKtcjJARkSRVnk5IAMieIrcnJAhkSRFTk5IEOimIqcHJAhUTRFTg7IkCiOIicHZEgUQZGTAzIkip3IyQEZEkVN5OSADIniJXJyQIZEkRI5OSBDohiJnByQIVF0RE4OyJAoLiInB2RIFBGRkwMyJIqFSC0iomXa1KMWx4BGcRA5uSBDogiInFyQIVHsQ04uyJAo6iEnF2TI5UzQmJUllXMGSe4/koz3uufMA3m6+FxvskTats5slzn7VXvdeKdPQassd17bcmdDy/pkFFnrF0q/oyhrmlqS0KVfF+JXLO9Pwg1aD8Vb0/qSW+m1Keua2pp3MYH0U21frV+3TsdLnq+a0ZJab+tQxt5kq6SabTX7VVLrbYV8MtIWejKj+i3fyxX3ny/P0j7X3lHbzOm510/xIdfsNPQyFk9WWe+8ofULuwIL+82r9Sb99ewqbdJfuaJv2t+m4ytX/f/tz5edGdhzl2/VrbB3yGLlzJu88myFyIWd0JYSeXoYNhQFcRUXt9H+b6uDpTtH73uTbcgk5tDetuZkS2ZgJO8Z9ewcSBwxnV8vNW+578jOdy4nXf2+s9rLiLXeMDNLRaqes7gnYzDqo/dWIefECGQtzLMx4Wt9h69ecVvF4l3a4KdhjJ96leq5i8Fr2xlU7JgnX2rjoEzIxal8fUT+h7lfum6gFn9BsO7+T8SXmTNTLb67W3Nv9Xv4Ft/drY2eaH+01rKN/dCrmMh49V48sHuxOg5P7rBd+Tzg369j3XBVy1W77n4u+0D7ulxjvY/KU9hJbaeN+Kp+DoeW0X08m9vSKplJrtaTa27rvr1r9dVH1rOsofV90YZYroZ83jY7+YmjZeXeGsiemVbKVQ860kbnEauq0Bc7OkP0HOzwRdSZyiqYmVnUnjkKo63vV85HdTsZtFHPAOq3pLmcmEIZ+9TWoR4NN1eos1vb++7f7dQLxwd399+9+5unB9sfvGf/puct9Sv1a75b9tVv1Qccr3W4v96Nf229vfWLrV+2/9L+W/sf7X8adOuGbfNztfTT/vf/AHXmu08=</latexit> • ? : (H, k · k H ) ! (C(X), k · k1) bounded linear; <latexit sha1_base64="AAhUkJm2ilTZM5Hud9w32OpW9KM=">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</latexit> h µ , hi H = Z X h'x , hi H dµ(x) = hµ , ?hiM,C
7. DATA-GAIA 23 Y. De Castro (ICJ) 3 Measure embedding <latexit sha1_base64="aJDnvLFL3xtlCZZ2ao5TCU5MSAw=">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</latexit> : µ 2 M(X) 7! Z X 'xdµ(x) <latexit sha1_base64="itENhbte+jL/xZD8T9WKnWjBpDY=">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</latexit> Assumption: ' is continuous <latexit sha1_base64="MkdnN6yVf+C8sTrSTta2POhoYmE=">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</latexit> • Kernel: K(x, x0) := h'x , 'x0 i H ; <latexit sha1_base64="dI+1oLveTiCfe40gg3jlkIFnma8=">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</latexit> • Well-defined and bounded linear; <latexit sha1_base64="5uBeJvWNeGWvmYts+24cKf4FS8A=">AAAll3icpZrrbhvHFcdHadOkbtrY7aeiXxYVCdiAakgCekEapKlvcRI7om1ZUpI1hD3kklxrL8zuUpcQah8wT5A3aN6iZ87MksvZP7l0LEHScOY3Z+7/nXNWNImjotzd/XHrnV/88t1fvff+r2/85oPf/u7Dm7d+f1Rk07wfvuxncZafUFCEcZSGL8uojMOTSR4GCcXhMZ3d1+XH52FeRFl6WF5NwldJMEqjYdQPSs46vfmTn2ZROgjT8ka349M0jsOy4x2HcfyXQThkowMvSAceZVOGBp5uJsj/eaNCfc8fZnkQx97Yj1I/CcoxkffY39lZlPjJtCrrB7H39PY8eXKHwY5ueBAVkzi4KsqrOPR748jf+cjfWVsxCSZFmXkMlKezRcG1fx7kk3F0eil5eeIN2Mztyzud05vbu3d35ctrJvZsYlvZr15269YPylcDlam+mqpEhSpVJadjFaiCv79Ve2pXTTjvlZpxXs6pSMpDda1ucN0pUyETAeee8e8Rf/rW5qb8WdsspHafW4n5J+eanupynYy5nNO6NU/Kp2JZ566yPRObum9X/JesrYRzSzXm3LZ6FblpPT2WUg3VP2QMEY9pIjl6dH1rZSqzonvu1UZVsoUJ5+n0gMtzTvelZjXPntQpZOx6bgMp/5+QOld/7lt2qn5a28tCnXNqecYvl+Z89cwMOHfIlG5nHZfKbBGPhSxn0jH/zpjJZcRjXpmCuSsZ3UzyAmHO5rsm5dSFzHwovdNzMnM+X3POoZPTlZGaeYykx6/Ynv7u2lmM7ayWMtO5nbMBp4f8t6/+zVb9+bqbGjPOvbY9W5D3IHkPkPcheR+QDyD5AJAPIfkQkI8g+QiQn0HyM0A+huRjQH4Oyc8B+QUkvwDkl5D8EpBPIPkEkE8h+RSQX0HyK0AeQPIAkD1I9gD5DJLPAPkcks8B+QKSLwB5CMlDQL6E5EtAHkHyCJDHkDwG5AkkTwD5NSS/BuQ3kPzGkkZhSPR00NAYT91mzZzM9b4vz89Q3Vlqg6z6VFaqZ1i2QoPIahDmm0pEVokw39QjsnqE+aYqkVUlzDe1iaw2Yb6pUGQVCvNNnSKrU5hvqhVZtcJ8U7PIahbmm8pFVrkw39QvsvqF+aaKkVUxzDe1jKyWYb6paGQVDfNNXSOra5hvqhtZdcN8U+PIahzmm0pHVukw39Q7snqH+abqkVU9zDe1j6z2Yb6pgGQVEPNNHSSrg5hvqiFZNcR8UxPJaiLmF8rYro36DnrRoo3Bkv4S5+q7Y7NPBDgCXB9wfcANADcAXAi4EPZvCMghIEeAGwFuDLgx4CLARYB7DbjXgDsD3BngYsDFgEsAlwAuBVwKuAxwGeAmgJsA7jvAfQe4HHA54ArAFYArAVcCbgq4KeDOAXcOuAvAXQDuEnCXgLsC3BXgvgfc9yvuWSPx/ULed8ivM36k9nITsZfa00xye9O+qFaQ2UoV85e46xabJO23WVxQbfZG4q0nrRbrXJvNnMeStdirmDZbsbSrFbKth8tkm91Edu06e0ltX6+2cynqts7OZU3/VtsxcYf2la1zq22ae0jEq5bIM221vQXVZu9AojqjFnsLqs3e4Xwk6+wdbjzeYqPxFhuPNxTNXnfGzG3IE7V4LXUr5aC1ESBdvhwDGohOI++LnBiQIZHfRU4MyJDI4yInBmRI5GuREwMyJPKyyIkBGRL5V+TEgAyJPCtyYkCGRD4VOTEgQyJvipwYkCGRH0VODMiQyIMiJwZkSOQ7kRMDMiTymsiJARkS+UvkxIAMiTwlcmJAhkQ+EjkxIEMi74icGJAhkV9ETgzIkMgjIicGZEjkC5FaeETLtMlHNY4AjfwgcmJBhkQeEDmxIEMi34ecWJAhkddDTizIkMuRoAkrSyb3DJLYfyQR73XPmYfydPE530SJtG0d2a5i9qvOuumdvgWtstx7Y8u9DS3rm1FkrZ8r/Y6iymmrSUJX/TqXfsXy/iTcoPZYemtqX3AtvTZVXltd8y5mIO3U69fz163T0VLPV81oRa23dV/G3marotpttferotbbCvlmpC30ZUb1W75XK/afL8/SIefeVtvM6bnXT/Ex59xpaWUiPVllvfeW1s/tCizst6/W27TXt6u0SXvVir5te5uOr1r1n9ueLydzYO9dvlW30u6QxcqZN3nV3QqRCzuhTaXy9DBsKAriKi6uo/u/rfaXdo4+9ybakIvPoXvbmZMdmYFE3jPq2dkXP2I2L680b7ntyM53ITdd/b6z3krCWm+Ya0tFqhmzuCdjMOqjz1Yp98QIRC3MszHlsqHD10vcWrH0Lmvpp2FMP/UqNWMXoze2M6rZMU++zPpBuZCLW/l6j/zjeb903kgt/oNg3f5PpS/Xzkx1eHd35r3V7+E7vLs7Gz3RPrHW8o37oVcxlfHqs7hvz2J9HJ7ssB35u8+/38S64eqW63bd81y1gc51tcb6HFW3sOPGSUu4VD+HQ8voNp7PbWmVzCVW60mZW3tod60ufWJ7lrfUfiDaEEtpyPdtc5IPHC2rztZIzsyslq73oCd1dByxrgpDsaMjRC/ACV94nZmsgpmZRe6pozDa+l7tftS0k0MbzQigfktayI0plLHPbB5q0XBzhTq9ub3n/t9OM3G0f3fvb3f/+mx/+9OP7P/0vK/+pP7Mu2VP/V19yv5aj9vrbx1sTbf+s/Xf7h+7/+o+6j426Dtbts4f1NJX99n/AZy4gVc=</latexit> • 8h 2 H , 8µ 2 M(X) , <latexit sha1_base64="aQyWDgWJ71C3fvl1NWXHw8/QAIM=">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</latexit> • ? : (H, k · k H ) ! (C(X), k · k1) bounded linear; <latexit sha1_base64="AAhUkJm2ilTZM5Hud9w32OpW9KM=">AAAl83icpZpbc9vGFcdXadqmSi92+5gXTETO2DOqR9JM007bZFLLTpzEjmlblpQEjgaHAElYIMAAoC7h8JPkrdO3Tj9JnvLY9gu036Jnzy54WfxJ0LU8FpcHvz17w/6x50A0SuKi3Nv759YbP3rzxz/56Vs/237757/45a9u3Pz1cZGN8270vJslWX5KQRElcRo9L+MyiU5HeRQMKYlO6PxQXz+5iPIiztKj8noUvRgG/TTuxd2gZNPZza17fprFaRil5Xa75dM4SaKy5Z1ESfLbMOqx19AL0tCjbMxQ6Ol2gvxPc9b3/F6WB0niDfw49YdBOSDyHvi7u/Mr/nBcXesGiffo1qx4epvB1nbLD+NilATXRXmdRH4SpH3+6AxiXZM97Q78XGxnk1kL0+33PfZaWpN4m1ZVL4J8NIjPrlZW9nelnA+9kNu4dXV7+/2qrm1SN/+1X5RB7jkOZAy7s+LhtHV2Y2fvzp78ePXCvi3sKPvTyW7e/F75KlSZ6qqxGqpIparkcqICVfC/r9S+2lMjtr1QE7blXIrleqSmapvrjpmKmAjYes6/+/ztK2tN+bv2WUjtLreS8P+ca3qqzXUy5nIu69Y8uT4Wz9q6yvdEfOq+XfMnWV9DtpZqwNamehW5aT09llL11B9kDDGPaSQWPbqu9TKWWdE99xZGVbKHEdt0OeTrOZe7UrOaZ0/qFDJ2PbeBXP+PkNqqv3ctO1b/XdvLQl1waXnGr5bmfPXMhGztMaXbWcelMlvEYyHLmXLCvzNmchnxgFemYO5aRjcRWyDM+eyuSbl0KTMfSe/0nEyc71O2HDmWtozUzGMsPX7B/vS/tp3FxM5qKTOd2zkLudzjz676C3v1Z+tuakzYOrU9m5N3IXkXkIeQPATkPUjeA+R9SN4H5EeQ/AiQH0PyY0A+gOQDQH4CyU8A+SkkPwXkZ5D8DJAPIfkQkI8g+QiQn0Pyc0A+huRjQHYg2QHkE0g+AeRTSD4F5DNIPgPkESSPAPkcks8BeQzJY0CeQPIEkKeQPAXkF5D8ApBfQvJLSxqFIdHTsKYxnrrFmjma6X1Xnp+Rur3UBln1qbxUz7BshQaR1SDM15WIrBJhvq5HZPUI83VVIqtKmK9rE1ltwnxdocgqFObrOkVWpzBfVyuyaoX5umaR1SzM15WLrHJhvq5fZPUL83UVI6timK9rGVktw3xd0cgqGubrukZW1zBfVzey6ob5usaR1TjM15WOrNJhvq53ZPUO83XVI6t6mK9rH1ntw3xdAckqIObrOkhWBzFfV0Oyaoj5uiaS1UTMz5WxWRv1GfSyQRuDJf0ltuqzY71PBDgCXBdwXcCFgAsBFwEugv3rAbIHyD7g+oAbAG4AuBhwMeBeAu4l4M4Bdw64BHAJ4IaAGwIuBVwKuAxwGeBGgBsB7hvAfQO4HHA54ArAFYArAVcCbgy4MeAuAHcBuEvAXQLuCnBXgLsG3DXgvgXctyvOWX2J/SK+71BcZ+JIHeUOxV9qdzPJ6U3HolpBJitVzF/ipg0+Sdpv8jinmvz1JVofNnpc5Jp85jyWrMFfxTT5SqRdrZBNPVwmm/wO5a5d52+4cF+v9nMl6rbOz9WC/q32Y/IOzSu7yK32ac4hMa/aUJ5pq/3NqSZ/jyWr02/wN6ea/B3NRrLO39HG4y02Gm+x8Xgj0ex1e8ychjxRi5dSt1IOWpsB0teXc0Ch6DSKvsjJARkSxV3k5IAMiSIucnJAhkSxFjk5IEOiKIucHJAhUXxFTg7IkCiyIicHZEgUU5GTAzIkiqbIyQEZEsVR5OSADIkiKHJyQIZEsRM5OSBDoqiJnByQIVG8RE4OyJAoUiInB2RIFCORkwMyJIqOyMkBGRLFReTkgAyJIiJyckCGRLEQqXlEtEwbO6pxDGgUB5GTCzIkioDIyQUZEsU+5OSCDImiHnJyQYZczgSNWFkyOWeQ5P5jyXive87cl6eLz3aTJdK+dWa7ytmv2uumd/oUtMpz55U9dzb0rE9GsfV+ofQ7isrSVJOErvp1If1K5P1JtEHtgfTW1L7kWnptKltTXfMuJpR2Fusv2tet0/FSz1fNaEWt93UoY2/yVVHNvpr7VVHrfUV8MtIeujKj+i3fixX3ny/P0h5bb6kd5vTc66f4gC23G1oZSU9Wee+8pvcLuwJz/82r9Trtde0qbdJetaKv296m46tW/f9tz5edGdpzl2/VrbR3yHzlzJu86myFyLmfyJZSeXoYNhIFcRUX19H931EHS3eO3vcm25BLzKF725qRLZmBobxn1LNzIHHEZHa90rzltmM734WcdPX7zsVWhqz1hplaKlb1nMVdGYNRH723SjknxiBrYZ6NKV/rOfziFbdWIr3LGvppGNNPvUr13EX/lf30F/yYJ19m46BcyPmpfH1E/udZv7Str+Z/QbDu/k+lL1Nnplp8d7dmvdXv4Vt8d7c2eqJ9YL3lG/dDr2Iq49V78cDuxcVxeHKH7crnAf9+Fe+GW/S86Nfdz1UbaF9Xa6z3UXUKO6nttCFf1c/hyDK6jaczX1olc8nVenLNrd2zd62++tD2LG+ofU+0IZGrEZ+3zU5+7GhZtbf6smcmC+XFHnSkjs4jLqpCT/zoDNEzsMPnUWcmq2BmZm49cxRGe99fOB/V/eTQRz0DqN+SFnJiimTsE2tDLRpuplBnN3b23b/bqReOD+7sv3fnd08Odj78o/2bnrfUO+pdvlv21e/Vhxyvdbi97tbft37Y+tfWv9vj9nftv7b/ZtA3tmyd36iln/Y//gfLpqTv</latexit> h µ , hi H = Z X h'x , hi H dµ(x) = hµ , ?hiM,C
8. DATA-GAIA 23 Y. De Castro (ICJ) Framework 4 <latexit sha1_base64="P6lfohRF8JNzmoHwPTiF/3GNg48=">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</latexit> • Many instances of ML problems can be formulated as an inverse problem Recover µı from Y ≥ P µı where : µ œ M ‘æ H with H a Hilbert space and M the space of signed regular Borel measures on some metric space X. <latexit sha1_base64="t7KTuzNMYUS6DvJSM8ecL04kwFM=">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</latexit> • The sought after target measure µı can be described by few parameters µı ƒ Kı ÿ k=1 aı k ”xı k namely, it is a discrete measure with aı k œ R and xı k œ X.
9. DATA-GAIA 23 Y. De Castro (ICJ) Some inverse problems 5 <latexit sha1_base64="BcTaQrT9UALdouA1Unx4RLCvIHQ=">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</latexit> Consider a metric space (X, dX ) of predictors and a space Y of observations. Given n observations Yn := {y1, . . . , yn } 2 Y⌦n, we would like to recover a tar- get µ? in M(X), the space of signed measures on X, from these n observations. <latexit sha1_base64="O0NpgCD+4IH7y54tD3h5OgyevMs=">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</latexit> Assume that the target is discrete µ? = K? X k=1 a? k x? k and we have 3 scenarii: Yn i.i.d µ?, (S1 : Sampling) Y = µ?, (n = 1) (S2 : Functional inference) Yn = nµ? + en , (S3 : Noisy linear measurements) where ak 2 R, x? k 2 X, is linear from M(X) to (probability) distributions (S1 and S2), n is linear from M(X) to Y⌦n (S3), and en some noise.
10. DATA-GAIA 23 Y. De Castro (ICJ) Some inverse problems 5 <latexit sha1_base64="BcTaQrT9UALdouA1Unx4RLCvIHQ=">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</latexit> Consider a metric space (X, dX ) of predictors and a space Y of observations. Given n observations Yn := {y1, . . . , yn } 2 Y⌦n, we would like to recover a tar- get µ? in M(X), the space of signed measures on X, from these n observations. <latexit sha1_base64="O0NpgCD+4IH7y54tD3h5OgyevMs=">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</latexit> Assume that the target is discrete µ? = K? X k=1 a? k x? k and we have 3 scenarii: Yn i.i.d µ?, (S1 : Sampling) Y = µ?, (n = 1) (S2 : Functional inference) Yn = nµ? + en , (S3 : Noisy linear measurements) where ak 2 R, x? k 2 X, is linear from M(X) to (probability) distributions (S1 and S2), n is linear from M(X) to Y⌦n (S3), and en some noise. <latexit sha1_base64="gmwHebuBZCprnzZYONC+UYLknpc=">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</latexit> µ? ! nµ? ! nµ? + en ! ˆ µn (S3)
11. DATA-GAIA 23 Y. De Castro (ICJ) Some inverse problems 6 <latexit sha1_base64="rDbmtk9MB1+Z4S6sX07ktQurFKE=">AAAFWHicjVLtahNBFL2JjW3Xr7b+9M9gV2ihhGwQlUCh4B9BhIi2DWTSMLs7yQ7ZL2dm88GSvqBPoE+gb+GdyaZaq9UJSe6ce+6ZOfeOn8dC6VbrS61+Z6Nxd3Nr27l3/8HDRzu7e2cqK2TAT4MszmTPZ4rHIuWnWuiY93LJWeLH/NyfvDb58ymXSmTpR73I+SBh41SMRMA0QsPd2me3pP6I9LveYOmSg3dirgvJ1WGHnAsdERXwlEmREffD0HOPiI44yXzF5dQKEHcxnLhEKKJYksc8JCOZJYSRZKVDQp4qoReX7uiCKs0kSrA0JBiNub50aVJUuNGwZSIdk5jNyCziWO+yVd6cwnC/1p1xMY60smLu/M+cnEmWcI3uOw7tO9UNyDFVRTIsJ8fe8qJ8uwKXV8fQKZN5JIblleiSHh3RTwULiUM1n+sSz1w6FnFownSE7aPdSKCXzjEVqR6WFg5YTHrLtd7cYjIhIfIO5ocOqjp04FQ2r2iuaVmmuelGZUCKYN1HMirSwDZ+Zqbz0yE2wW0Od/ZbzZZd5GbgVcE+VKub7dY6QCGEDAIoIAEOKWiMY2Cg8NMHD1qQIzaAEjGJkbB5DktwsLZAFkcGQ3SCv2Pc9Ss0xb3RVLY6wFNi/EqsJPAMazLkSYzNacTmC6ts0L9pl1bT3G2B/36llSCqIUL0X3Vr5v/WGS8aRvDKehDoKbeIcRdUKoXtirk5+cWVRoUcMROHmJcYB7Zy3Wdia5T1bnrLbP6bZRrU7IOKW8D3W90Z1mqC0nq6bT4l3kehp8wqr3iGpe19UuSYnlBbJa3nsLpFWb2G22avYIrR9dnPr03fwXfq/f4qbwZn7ab3ovn8fXv/pFO92C14Ak/hAF/lSziBN9CFUwjq7Xqvzur+xtcGNDYb2ytqvVbVPIZrq7H3A5qYWtc=</latexit> [P1] (Mixtures): With scenario S1, the observation yk is sampled from a mixture density f?, and target µ? is a mixing law where a? k are mixture weights and x? k are mixture parameters: f? = K? X k=1 a? k 'x? k , and µ := Z X 'xdµ(x) , where 'x denotes a parametric density function with parameters x.
12. DATA-GAIA 23 Y. De Castro (ICJ) Some inverse problems 6 <latexit sha1_base64="rDbmtk9MB1+Z4S6sX07ktQurFKE=">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</latexit> [P1] (Mixtures): With scenario S1, the observation yk is sampled from a mixture density f?, and target µ? is a mixing law where a? k are mixture weights and x? k are mixture parameters: f? = K? X k=1 a? k 'x? k , and µ := Z X 'xdµ(x) , where 'x denotes a parametric density function with parameters x. <latexit sha1_base64="M134ZZ9zsJf6tUf5gnQbbF9v3Bs=">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</latexit> [P2] (Continuous Sparse): With scenario S3, the observation yk is a noisy linear measurement of µ?: yk = ( nµ?)k + ek and ( nµ)k := Z X kdµ , (1) where k is some known bounded function.
13. DATA-GAIA 23 Y. De Castro (ICJ) 7 Interlude: Super-Resolution changing the rates of photoactivation and photobleaching, even if the density of fluorophore-labeled molecules is much higher than one fluorophore per mm2. We present a novel method by which fluorescence micros- copy may be performed to obtain an image with greatly enhanced ability to resolve large numbers of fluorescent 1. The spontaneous interconversion activated (fluorescent) state mus light-controlled activation rate. 2. For irreversible photoactivatio quantum yield must be finite a FIGURE 1 localization area contain (here, PA- neously wit for readout spatial illum a second on diode laser, Within the vation beam blue circles circles) and time, the ac (red Xs) an (black circl then activated, localized, and bleached until a sufficient number of molecules have been analyzed to construct an image. (G) Th the 405-nm activation laser (X405), which is reflected by a dichroic (DM1) to make it collinear with the Ar1 readout laser. A inverted fluorescence microscope is used to focus the lasers, which are reflected upward by a second dichroic mirror (DM2 objective lens (OBJ). The sample, supported by a coverslip (CS), emits fluorescence which is collected by the objective, transm and focused by the tube lens (TL) to form an image on a camera (CCD). Biophysica S. Hess, T. Girirajan, M. Mason, Ultra-High Resolution Imaging by Fluorescence Photoactivation Localization Microscopy, Biophysical Journal (2004).
14. DATA-GAIA 23 Y. De Castro (ICJ) 8 Interlude: Super-Resolution Y. Li, S. Osher, R. Tsai, Heat Source Identi fi cation based on L1 Constrained Minimization, Inverse Problems and Imaging (2014). 210 Yingying Li, Stanley Osher and Richard Tsai (a) Heat source u0 (b) Au0 = f (c) f + noise
15. DATA-GAIA 23 Y. De Castro (ICJ) 9 Interlude: Super-Resolution H. Pan, T. Blu, M. Vetterli, Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy, IEEE trans. on Signal Processing (2017). EE, Thierry Blu, Fellow, IEEE, and Martin Vetterli, Fellow, IEEE continuous-time arrival problem, rliest occurrence Prony’s method. n high resolution -time sparse sig- sampling theory ectral estimation pling. But not all fied by a concrete s, we develop the , typically sum of his by identifying uniform samples uniform samples A valid solution ization such that en measurements Fig. 1. Schematic diagram of a radio interferometer. The cross-corre of the received signals at different antennas are related to the Fourier tra of the sky image (see Table I) at certain non-uniform frequencies.
16. DATA-GAIA 23 Y. De Castro (ICJ) 10 Interlude: Super-Resolution Perfect Recovery Discrete Measure <latexit 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µ? <latexit 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<latexit 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min µ : (µ)=Y kµk1 <latexit 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Y = (µ?)
17. DATA-GAIA 23 Y. De Castro (ICJ) Some inverse problems 11 <latexit sha1_base64="kvdNaI0VPo6MqUSahvgrqderNe4=">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</latexit> [P3] (Two-layer neural networks): With scenario S3, one observes a couple (yk, zk) of input data zk and response yk as a linear measurement of µ?: f? = K? X k=1 a? k 'x? k , yk = ( nµ?)k + ek and ( nµ)k := Z X 'x(zk)dµ(x) , where f? is the target function, 'x(z) := (hx, (1, z)i) = (x1 + Pd j=2 xjzj) is the neuron outcome with activation and weights x at input point z.
18. DATA-GAIA 23 Y. De Castro (ICJ) Some inverse problems 11 <latexit sha1_base64="kvdNaI0VPo6MqUSahvgrqderNe4=">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</latexit> [P3] (Two-layer neural networks): With scenario S3, one observes a couple (yk, zk) of input data zk and response yk as a linear measurement of µ?: f? = K? X k=1 a? k 'x? k , yk = ( nµ?)k + ek and ( nµ)k := Z X 'x(zk)dµ(x) , where f? is the target function, 'x(z) := (hx, (1, z)i) = (x1 + Pd j=2 xjzj) is the neuron outcome with activation and weights x at input point z. <latexit sha1_base64="Z9gMox+zgoYJPkyd8jz8JDlZmqA=">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</latexit> [P4] (Kernel Sparse Designs): With S2, given f?, we aim at µ? s.t. F(µ) := µ Z X 'x(·)f?(x)dx F , with µ := Z X 'xdµ(x) , is the best approximation (for the criterion F(µ)) of f?, and 'x feature map at feature input point x 2 X.
19. DATA-GAIA 23 Y. De Castro (ICJ) BLASSO 12 <latexit sha1_base64="DRjIGdxUkUokfYj55AvDsBAsdRk=">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</latexit> [P5] (Symmetric Tensors): With scenario S3, we aim at finding a K?-rank d-way symmetric tensor ? from the noisy linear measurements yk: ? := K? X k=1 a? k x? k ⌦d = Z X x⌦ddµ?(x) , ( nµ)k := Z X k(x)dµ(x) , with a? k > 0, x? k are normalized (kx? k k2 = 1), k(x) := h⌧k , x⌦di for some ⌧? k 2 (Rn)⌦d (e.g., the canonical basis), h· , ·i the standard dot product of tensors, and ek 2 R some centered random variable. Note that ( nµ?)k = h⌧k , ?i is some linear form evaluated at the target point ?.
20. DATA-GAIA 23 Y. De Castro (ICJ) BLASSO 12 <latexit sha1_base64="DRjIGdxUkUokfYj55AvDsBAsdRk=">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</latexit> [P5] (Symmetric Tensors): With scenario S3, we aim at finding a K?-rank d-way symmetric tensor ? from the noisy linear measurements yk: ? := K? X k=1 a? k x? k ⌦d = Z X x⌦ddµ?(x) , ( nµ)k := Z X k(x)dµ(x) , with a? k > 0, x? k are normalized (kx? k k2 = 1), k(x) := h⌧k , x⌦di for some ⌧? k 2 (Rn)⌦d (e.g., the canonical basis), h· , ·i the standard dot product of tensors, and ek 2 R some centered random variable. Note that ( nµ?)k = h⌧k , ?i is some linear form evaluated at the target point ?. <latexit sha1_base64="GECKhd4GeikVHMbseN7NpZhlxbM=">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</latexit> We consider BLASSO solutions: ˆ µn œ arg min µœM(X) ) Fn(Yn, µ) + ⁄n |µ| * where the total variation norm is |µ| = sup ) s X fdµ : |f| Æ 1 * , the so-called “ data fitting ” term Fn(Yn, µ) quantifies how much the measure µ is likely to fit the data Yn (here, the smaller the better), and ⁄n > 0 a tuning parameter.
21. DATA-GAIA 23 Y. De Castro (ICJ) 13 Interlude: Regularization path on Tensor PCA <latexit sha1_base64="ESDm+djQ8vbl7YqhsdkMg6BSjH4=">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</latexit> x 7! hY, 'x i
22. DATA-GAIA 23 Y. De Castro (ICJ) 13 Interlude: Regularization path on Tensor PCA <latexit sha1_base64="ESDm+djQ8vbl7YqhsdkMg6BSjH4=">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</latexit> x 7! hY, 'x i <latexit sha1_base64="6h8eUuqXsrP7DDu6zja3IGCe4Eg=">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</latexit> x 7! hY b a1b x⌦d 1 ,'x i 1 hx,b x1 id
23. DATA-GAIA 23 Y. De Castro (ICJ) The Mixture program 14 <latexit sha1_base64="rDbmtk9MB1+Z4S6sX07ktQurFKE=">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</latexit> [P1] (Mixtures): With scenario S1, the observation yk is sampled from a mixture density f?, and target µ? is a mixing law where a? k are mixture weights and x? k are mixture parameters: f? = K? X k=1 a? k 'x? k , and µ := Z X 'xdµ(x) , where 'x denotes a parametric density function with parameters x.
24. DATA-GAIA 23 Y. De Castro (ICJ) The Mixture program 14 <latexit sha1_base64="rDbmtk9MB1+Z4S6sX07ktQurFKE=">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</latexit> [P1] (Mixtures): With scenario S1, the observation yk is sampled from a mixture density f?, and target µ? is a mixing law where a? k are mixture weights and x? k are mixture parameters: f? = K? X k=1 a? k 'x? k , and µ := Z X 'xdµ(x) , where 'x denotes a parametric density function with parameters x. • MLE is a non-convex program • Usually solved by EM • K is assumed to be known • Convergence of EM to global maximum can be tedious <latexit sha1_base64="BVLgunTX85W28llXDg5PV2q6OaM=">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</latexit> Log-Likelihood at ◊ = (a1, . . . , aK, x1, . . . , xK ): arg min ◊ Ó ≠ n ÿ i=1 log ! K ÿ k=1 akÏxk (yk ) "Ô
25. DATA-GAIA 23 Y. De Castro (ICJ) 15 <latexit sha1_base64="MiHMKuCYUs9SOpdRRqyj4v9LGxE=">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</latexit> • Empirical measure: ˆ fn = 1 n n ÿ i=1 ”yi • PDF: µı = K ÿ k=1 aı k Ï(xı k ≠ ·) • Inverse problem: µı æ µı æ ˆ fn Solving the Mixture: Hilbert space for data fi delity
26. DATA-GAIA 23 Y. De Castro (ICJ) 15 <latexit sha1_base64="MiHMKuCYUs9SOpdRRqyj4v9LGxE=">AAAEsnicjVLtbtMwFL0ZAUb5WuEnfyI6pCFE1UwTQ5MqTeJDIP4UiX2IeYucxG2s2klkO2Wl6kvwNPyFt+AN4C24dtNpZaOQqsnxueee63vtuBRcm07nh7dyxb967frqjcbNW7fv3F1r3tvXRaUStpcUolCHMdVM8JztGW4EOywVozIW7CAevrDxgxFTmhf5BzMu2bGkg5z3eUINUlHTe0JiNuD5hBsm+Wc2bRCLgley5ApVIpCM6kqxHUIa6yTluhR0rM1YsIBk1AT9KO+SvqJJmBNdyWjCu+H0JCcpE4ZGk3HEp+u1Z+/l68tcJDVZ3A9IL+NEVidEG6q6M6+h9XpHZ1w0JCOqyoxvnEbDGfWUJGlhHs8LvM1tqywoVYH9y1mxuSVRfJDhVxWfLi15Ln7WGRqzPD2bTbTW6rQ77gkugrAGLaifXtH0toFACgUkUIEEBjkYxAIoaPwdQQgdKJE7hglyChF3cQZTaGBuhSqGCorsEN8DXB3VbI5r66lddoJVBP4VZgbwCHMK1CnEtlrg4pVztuzfvCfO0+5tjN+49pLIGsiQ/VfeXPm/ebYXA3147nrg2FPpGNtdUrtUbip258G5rgw6lMhZnGJcIU5c5nzOgcvRrnc7W+riP53Ssnad1NoKfi3tLkW2j7WtdtnJaBghWjyZ04WzWTZBgciquJuadHdhWYZATYF7s1PIrRJvaPjnfbwI9jfb4bP21vvN1u5OfVdX4QE8hA28j9uwC2+gB3uQeF+8r94377u/5X/0qZ/MpCtenXMfFh5f/AbikiX2</latexit> • Empirical measure: ˆ fn = 1 n n ÿ i=1 ”yi • PDF: µı = K ÿ k=1 aı k Ï(xı k ≠ ·) • Inverse problem: µı æ µı æ ˆ fn <latexit sha1_base64="X3h4wR49MjjcuKavIL2YzSkCzHA=">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</latexit> • Convolution operator L(f) = ? f, where is such that F[ ](t) = 1{|t|T } with F the Fourier transform. • Kernel defines a RKHS L <latexit sha1_base64="QIRCQoDVQB+x6okIctKE3iKHVUA=">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</latexit> <latexit sha1_base64="jmbsFTv5ZHoWQ4CG7VqaOLAZjuQ=">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</latexit> L = f : Rd ! R s.t. kfk2 L = Z [ T,T ]d |F[f](t)|2dt < +1 Solving the Mixture: Hilbert space for data fi delity
27. DATA-GAIA 23 Y. De Castro (ICJ) 15 <latexit sha1_base64="MiHMKuCYUs9SOpdRRqyj4v9LGxE=">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</latexit> • Empirical measure: ˆ fn = 1 n n ÿ i=1 ”yi • PDF: µı = K ÿ k=1 aı k Ï(xı k ≠ ·) • Inverse problem: µı æ µı æ ˆ fn <latexit sha1_base64="X3h4wR49MjjcuKavIL2YzSkCzHA=">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</latexit> • Convolution operator L(f) = ? f, where is such that F[ ](t) = 1{|t|T } with F the Fourier transform. • Kernel defines a RKHS L <latexit sha1_base64="QIRCQoDVQB+x6okIctKE3iKHVUA=">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</latexit> <latexit sha1_base64="jmbsFTv5ZHoWQ4CG7VqaOLAZjuQ=">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</latexit> L = f : Rd ! R s.t. kfk2 L = Z [ T,T ]d |F[f](t)|2dt < +1 Solving the Mixture: Hilbert space for data fi delity Both in L <latexit sha1_base64="CeWjxSbR51SW1RytS6wEZUMkL7s=">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</latexit> <latexit sha1_base64="H8nQWVdnFOqUg2SKmDqSeISQJBc=">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</latexit> µ ≠æ Ï ı µ ≠æ Ï ı µ ≠æ T (µ) := ⁄ ı Ï ı µ <latexit sha1_base64="kZOAZ+Dkb9w/hKMyTksy+XSEtc4=">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</latexit> µı ≠æ Ï ı µı ≠æ ˆ fn ≠æ L( ˆ fn ) := ⁄ ı ˆ fn <latexit sha1_base64="OixdazuoycYhMa0ZCQb4SYNrg3w=">AAAD03icjVJdSxtBFL1x+2Hth7E+9mWpFCLYsBtKLUJBEEofLKRg1OLaZXYzawZnZ5fdWVFiXqSv/TX9Nb62L/YP9MEnz0w20FSadkKSM+eec2funRvlUpTa8y4bc86du/fuzz9YePjo8ZPF5tLT3TKripj34kxmxX7ESi6F4j0ttOT7ecFZGkm+Fx1vmfjeCS9KkakdfZbzw5QdKZGImGlQYfNDkDI9iBL3XahaE/wpVGtBWq2+DZKCxX4nON9uBQOm3SRUqy+DbinCnZYRBOfhcGyK3O3R507YXPHanl3ubeDXYGXT/3V9cTVa62ZLjXUKqE8ZxVRRSpwUaWBJjEp8Dsgnj3JwhzQEVwAJG+c0ogV4K6g4FAzsMX6PsDuoWYW9yVlad4xTJL4FnC69gCeDrgA2p7k2XtnMhv1b7qHNae52hv+ozpWC1TQA+y/fRPm/PlOLpoTe2BoEasotY6qL6yyV7Yq5uftbVRoZcnAG9xEvgGPrnPTZtZ7S1m56y2z8yioNa/Zxra3o58zq+mATnG20s16mpBOg6Zc5nXqbWR2UQEYlbNdSOwuzHBKaDHczXVBGiQn1/5zH22C30/Zft199xKhu0HjN0zN6Ti3M4zpt0nvqUg8nfaNL+k4/nJ4zdC6cL2PpXKP2LNPUcr7eACfY1iE=</latexit> Fn (Yn, µ) = 1 2 ÎL( ˆ fn ) ≠ T (µ)Î2 L
28. DATA-GAIA 23 Y. De Castro (ICJ) Solving the Mixture program: Basis Pursuit 16 <latexit sha1_base64="1VI0CH+BB7gRZgGWPDCmmFIooDQ=">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</latexit> • The TV-norm |µ|1 = sup Ó s X fdµ : 1 ≠ f Ø 0 & 1 + f Ø 0 Ô • Related to the space of nonnegative functions in a dual way.
29. DATA-GAIA 23 Y. De Castro (ICJ) Solving the Mixture program: Basis Pursuit 16 <latexit sha1_base64="iNyzvIOGxDyw2V6O8QKKDZ44hEI=">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</latexit> Beurling Minimal Extrapolation (BME) a.k.a. Basis Pursuit min µ Ó |µ|1 : T (µ) = T (µı) Ô (BME) <latexit sha1_base64="1VI0CH+BB7gRZgGWPDCmmFIooDQ=">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</latexit> • The TV-norm |µ|1 = sup Ó s X fdµ : 1 ≠ f Ø 0 & 1 + f Ø 0 Ô • Related to the space of nonnegative functions in a dual way.
30. DATA-GAIA 23 Y. De Castro (ICJ) Solving the Mixture program: Basis Pursuit 16 <latexit sha1_base64="iNyzvIOGxDyw2V6O8QKKDZ44hEI=">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</latexit> Beurling Minimal Extrapolation (BME) a.k.a. Basis Pursuit min µ Ó |µ|1 : T (µ) = T (µı) Ô (BME) <latexit sha1_base64="VMTRwKSazNinO2y/01JuwO8BCm4=">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</latexit> Theorem (A) ≈∆ (B) where (A): µı solution to (BME); (B): there exists P œ Range( € T ) such that (i) |P| Æ 1 & P(xı k ) = sign(aı k ) , ’k œ [K] and µı is the unique solution to (BME) if furthermore (ii) |P(x)| < 1 , ’x / œ {xı 1, . . . , xı K } . <latexit sha1_base64="1VI0CH+BB7gRZgGWPDCmmFIooDQ=">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</latexit> • The TV-norm |µ|1 = sup Ó s X fdµ : 1 ≠ f Ø 0 & 1 + f Ø 0 Ô • Related to the space of nonnegative functions in a dual way.
31. DATA-GAIA 23 Y. De Castro (ICJ) Alice and Bob 17 <latexit sha1_base64="wrwevCvnGhDGplZjSDp+DOrJYao=">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</latexit> • For all µı œ M ™ M(X), Alice sends T (µı) and Bob has to uncover µı; • Impossible for M = M(X); • Solved by (BME) decoder when M = M Ø C · K1/2 · d3/2 · # 1 T $ [≠T, T]d ™ Supp(F[Ï]) where M := Ó µ : ÷K Ø 1 , ÷x1, . . . , xK œ X s.t. min i”=j dX (xi, xj ) Ø Ô with X ™ Rd. <latexit sha1_base64="3Ipix8gf3cGb9wE4t6lecxXS9As=">AAADnXicjVJdSxtBFD1x22rtV2wffBDapVKwUMKmSC19EmpFSikKxghGwuxkooOzH+zOSiX4a/qqP6j/oD75FzwzrtBUTDshyZlzz7kz986Nc6NLG0W/GlPBvfsPpmcezj56/OTps+bc850yqwqpOjIzWbEbi1IZnaqO1dao3bxQIomN6sZHn128e6yKUmfptj3J1X4iDlI91FJYUv3m/FIvEfZQChNu9Efb73prylhx+rbfXIxakV/hbdCuweLqy/VRf+Hy62Y211hBDwNkkKiQQCGFJTYQKPnZQxsRcnL7GJEriLSPK5xilt6KKkWFIHvE3wPu9mo25d7lLL1b8hTDb0FniDf0ZNQVxO600Mcrn9mxd+Ue+Zzubif8j+tcCVmLQ7L/8t0o/9fnarEY4qOvQbOm3DOuOllnqXxX3M3DP6qyzJCTc3jAeEEsvfOmz6H3lL5211vh47+90rFuL2tthYuJ1Q3IDnm20056mRLHROMv82PsbSZ10BA5lfZdS/wsTHIYajLezXUhdUpOaPvvebwNdt632h9ay1sc1U+4XjNYwGsscR5XsIoNbKLjz/6JM5wHr4Ivwbfg+7V0qlF7XmBsBd0rEJK/ug==</latexit> (HT, )
32. DATA-GAIA 23 Y. De Castro (ICJ) Alice and Bob 17 <latexit sha1_base64="zNLsdx9WW74d9oAfbzFaiM4F3aY=">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</latexit> • What can be said about BLASSO? ˆ µn œ arg min µ Ó 1 2 ÎL( ˆ fn) ≠ T (µ)Î2 L + Ÿn |µ|1 Ô <latexit sha1_base64="wrwevCvnGhDGplZjSDp+DOrJYao=">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</latexit> • For all µı œ M ™ M(X), Alice sends T (µı) and Bob has to uncover µı; • Impossible for M = M(X); • Solved by (BME) decoder when M = M Ø C · K1/2 · d3/2 · # 1 T $ [≠T, T]d ™ Supp(F[Ï]) where M := Ó µ : ÷K Ø 1 , ÷x1, . . . , xK œ X s.t. min i”=j dX (xi, xj ) Ø Ô with X ™ Rd. <latexit sha1_base64="3Ipix8gf3cGb9wE4t6lecxXS9As=">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</latexit> (HT, )
33. DATA-GAIA 23 Y. De Castro (ICJ) Dual Certi fi cate of the BLASSO 18 <latexit sha1_base64="GTNTOG51j+E1fILa9tM2+cUwkwo=">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</latexit> Dual Certificate Under (HT, ), there exists P such that 1. P œ Range( € T ); 2. |P| Æ 1 & P(xı k ) = 1 , ’k œ [K], 3. Near Region: P(x) Æ 1≠CT2 Îx≠xı kÎ 2 2, for all x s.t. Îx≠xı kÎ2 . (1/T) 4. Far Region: P(x) Æ 1 ≠ ÷, for all x s.t. Îx ≠ xı kÎ2 & (1/T)
34. DATA-GAIA 23 Y. De Castro (ICJ) Exploiting the Bregman divergence 19 <latexit sha1_base64="vn+1IurrFk0ExfzGyC12qoxgmpM=">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</latexit> • P is a subgradient of the TV-norm at point µı; • Bregman divergence DP (µ, µı) := |µ|1 ≠ |µı|1 ≠ ⁄ X Pd(µ ≠ µı) Ø 0 ; • Taylor expansion: |µ|1 = |µı|1 + ÈP, µ ≠ µıÍ + DP (µ, µı)
35. DATA-GAIA 23 Y. De Castro (ICJ) Exploiting the Bregman divergence 19 <latexit sha1_base64="vn+1IurrFk0ExfzGyC12qoxgmpM=">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</latexit> • P is a subgradient of the TV-norm at point µı; • Bregman divergence DP (µ, µı) := |µ|1 ≠ |µı|1 ≠ ⁄ X Pd(µ ≠ µı) Ø 0 ; • Taylor expansion: |µ|1 = |µı|1 + ÈP, µ ≠ µıÍ + DP (µ, µı) <latexit sha1_base64="KaucstCv7PmRe5gw+snna/FVOQo=">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</latexit> Theorem Under (HT, ), for Ÿn = On (fln ) where E Ë ÎL( ˆ fn ) ≠ T (µı)Î2 L È . fl2 n , (Bound on the “noise”) it holds E Ë DP (ˆ µn, µı) È . fln
36. DATA-GAIA 23 Y. De Castro (ICJ) L1-Regularization for Mixtures 20 <latexit sha1_base64="/lLZGFWRvp3Pz8b8h01eVGxAXrY=">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</latexit> Theorem Under (HT, ), for Ÿn = On (fln ) where E Ë ÎL( ˆ fn ) ≠ T (µı)Î2 L È . fl2 n , (Bound on the “noise”) it holds T 2 Ë Kı ÿ k=1 |ˆ µn |(Nk )”xı k , |µı| È . fln , |ˆ µn | Ë X\ ! Kı € k=1 Nk "È Æ fln , and max kœ[Kı] |ˆ ak ≠aı k | . fln where T 2 is a partial transport distance close to unbalanced Wasserstein-2, Nk is the ball centered at xı k with radius Ô fln and ˆ ak = ˆ µn (Nk ). Furthermore, one can prove that fln = On (1/ Ô n) .
37. DATA-GAIA 23 Y. De Castro (ICJ) Optimization strategies for BLASSO 21 <latexit sha1_base64="gWT57j7NEYDBub1HllnLdJrZUsQ=">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</latexit> • Conic Particle Gradient Descent: optimizes a function on m parame- ters (xk, ak ) (called particles) that can be written as Fn (Yn, qm k=1 ak”xk ) then we can study the limit m æ Œ (over-parametrized) by considering the objective µ ‘æ Fn (Yn, µ) on the space of measures as in the works of Bach and Chizat, and Chizat. • Sliding Frank-Wolfe: solves convex programs on weakly compact sets (e.g., closed balls of the TV -norm for the weak-ı topology). This algorithm is a conditional gradient descent that may converge in a finite number of steps (Denoyelle, Duval, Peyr´ e, and Soubies) under some conditions. • Kernel SoS: Based on representation of nonnegative function based and a subsampling strategy, see Bach, Rudi, and Marteau-Ferey, and Lasserre, Magron, et al. • Other popular methods: Prony-type spectral methods such as MUSIC and ESPRIT, and non-convex approaches based on greedy minimization (e.g., (COMP) and “Continuous” LARS), see Elvira, Gribonval, Soussen, and Herzet.
38. DATA-GAIA 23 Y. De Castro (ICJ) 22 As of today’s research <latexit sha1_base64="lDvB+56WifCHiIHoF5UNc74a8tk=">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</latexit> • Sketching BLASSO to reduce the number of measurements up to the Infor- mation Theory limit; Ongoing with R. Gribonval and N. Jouvin <latexit sha1_base64="iTzU0oRdShpgxqnMKRuoiOg4a08=">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</latexit> • SGD Conic Particle Gradient Descent; Ongoing with S. Gadat and C. Marteau <latexit sha1_base64="LKTD4+kZmD/VRC8aQ9ysokdnXCg=">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</latexit> • Logistic BLASSO; Ongoing with Antoine SIMOES (PhD @ECL) <latexit sha1_base64="DATl+N3QO18/PM1qyRktT0JPg9Y=">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</latexit> • Inference BLASSO; Ongoing with F. Dalmao (Uruguay) and JM Aza¨ ıs <latexit sha1_base64="j0atb106uqHdS0wh+E0uDpDbVn4=">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</latexit> • Tensor BLASSO dual certificate; Ongoing with C. Boyer and V. Duval <latexit sha1_base64="P/LI/wg0OIHLyeIWoODXOapbdSM=">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</latexit> • Convolutional Kernel Networks, Dual certificates for 2 layers NN, Kernel Sparse designs, Kernel SoS... feel free to join!
39. DATA-GAIA 23 Y. De Castro (ICJ) End of the presentation… 23 Courtesy of Nicolas Jouvin (ex ICJ, now INRAE) https://nicolasjouvin.github.io/
40. DATA-GAIA 23 Y. De Castro (ICJ) End of the presentation… 23 Courtesy of Nicolas Jouvin (ex ICJ, now INRAE) https://nicolasjouvin.github.io/