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2022-09-14-epfl-ot-ml.pdf

Gabriel Peyré
September 14, 2022
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 2022-09-14-epfl-ot-ml.pdf

Gabriel Peyré

September 14, 2022
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  1. Optimal Transport for Machine Learning Gabriel Peyré É C O

    L E N O R M A L E S U P É R I E U R E
  2. Comparing Distributions for Learning <latexit sha1_base64="vT0xGll5ZYO8gmdwavYy0Ckm65g=">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</latexit> ATAC-seq

  3. 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="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Comparing Distributions for Learning <latexit 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<latexit 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<latexit 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↵✓ <latexit 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ATAC-seq
  4. min ✓ D(↵✓, ) <latexit 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<latexit 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<latexit 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Density fitting: <latexit 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<latexit 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<latexit 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<latexit 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Parametric model: ✓ 7! ↵✓ <latexit 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<latexit 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<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">AAA/qHiczVtbdxu3EUbSW6xe4rSPfei2shs7dX1kNeekbU6OYl0sK6Yt2RJ9SRirvKyojZdcencpW2aZP9PH/oT+gr62T/0H9U/oW2cGwAJLYnewbnpOhCMJC+KbGQyAuQDL3iSOsnxt7V9vvf2d737v+z9458LKD3/045+8e/G9nz7MkmnaD9v9JE7Sx71uFsbROGznUR6Hjydp2B314vBR79kWfv7oLEyzKBkf5eeT8MtRdziOTqJ+N4em44ufdPLwJeBmB920OwrzNOoHo2QQxn+cB5c6+WnQGXUnWZ4Esw4xm6XhYN7pxsfw2fzS8cXVtetr9BMsV26oyqpQPwfJe7/4QHTEQCSiL6ZiJEIxFjnUY9EVGZQvxA2xJibQ9qWYQVsKtYg+D8VcrAB2Cr1C6NGF1mfwdwhPX6jWMTwjzYzQfeASw28KyEBcBkwC/VKoI7eAPp8SZWytoj0jmijbOfzvKVojaM3FKbRyON3THzeA1hOQDTVTP+Yc+v2exhpB7wm1oBb6ituUtIcjDKzR50BhAm1YH8DnKdT7hNTzERAmIx3hHHTp839TT2zF577qOxWvaTTfFv19+/RyuUEJxJ64J7ZES7TFttgRhyRvfVkh3BaMYwI6TkHyIWgIpb0C0lyF/+uwp9bER1DbBRl71CeEsQTiAP6fk/wrStZA3BU3xZG4TbyvQP0u1K42mCeup17bGc2G/7pZYVZZSjNyXitBTuOfkgVIxcvavrh2Bmo2T2g9ZGpH1s+p1OCWeCD2a+cPOY+B/guyRCPSIHKcQfsQZMIVGgM/tIm4lv8qvharUP9a/E2t4wDkStXKPiV68smf5p+gIM33oczFBaLZBqlXilEG8CRnIClWf1ec0d4IyJJP4TO5l2Jae2PqVTe6e1DmVNOrpgdlRq3zWuQWFBdyi0XuQnEhd1lkC4oL2WKRB1BcyAOFrMPeh+LC3me5PoDiQj5gkUdQXMgjFvkQigv5kEV+DsWF/JxF3oLiQt5ikXeguJB3WGQbigvZZpE7UFzIHYWs3mto1xKiEzH76ibUyzzQT8XQcpOVb5NsrAu76bEr+xVYfl9uw383dttDp2EFdsdj9ZxUYPn1swtWzo3lrclt8pEu7G0WuwcrwI3dY7Gfia8qsJ957JdnFVh+x7TI37iwvP28C09u7F0Wew9qbizvZfahxY3d97D5kwrsAYu9L55XYH2sflqB5e3+IcViLuyhh8/IK7C812hD7ODG8vb0IcQgbizvcx5Bqxv7iMU+pqjRhX3MYp9QdOrCPvHwk68qsNpTyuh9SDFgCDu2jlq32JVYmwC1LsM/LnwL1tBHDVjMsMAMCTNiEbsFYtcT0SoQLW+5ssKOZhSx8lwOC8ShJ6JX+Cas5Wz/QdF/QLkZj9guENsLiLosAOdaj+WMogvdwiHzwnNhzWdMSWG/sRaq9VBveTViv4SQa/uUVv41ytUxkxlQ3lpN7bTw8RIZ0HMd4gXlenqUmgePywurYKNesqieA9VjUecO1DmLmjpQUxZ15kCdsSiz821cx2MFGP3jXMzoSef+3FlK+exiHzzuDng/bNmH/z5nKfVZOZ4IoKfU2bOxxSnUZpRJm8xum3JkedIQgmSy5746Y8InPJucqV0n7fC88OWB0Cee/nQikmdY0MF4MaAd1YzOHWqZU3wna83wt4udr2vN8Duk8TnF8bLWDJ8r6fM3kP1IYY/eAHsI+2mitG/qTWnIMxRJQ9c527xX2E08r3xJO0e2NeW/RTWpA1NvRiOzxpCVxtCEhtFlZumyGRWMkWRsq2tB45GMVXZr6k1lSMhXjpUc5qnpzGCfgZoZXW9G4wDiqi3KrGdWvekKnRSjMfVmNB4KebY+p3hd15vRGNKz1IepN6OBZypdlc2belPrjRqQGbKp65glpShGnzhH5L3rz1LsiHzZx+CJytMigq+nZCLPajq9ws/US7RoXZrIgd5/akVIZRozsc5mP1KGvOR7l+kY/4uab4EWA9i18lyfO5OOQUJ9YhDSKflTolaPKY9M49ZZHFqSkwVUR7XmbCxn+MoznXLbMbVyWZMZrdFjh+xsRmtvQhFbizTL6aFVOcNVFDkNtUoa4uk10d0r2sHJgvbXWNxkATEpVlqfbofkLXV9FunS+qGl48vqFiWHIu9UzPrFs+ATwqWUkSRkbVCWOp52P33KY7ehP7wmzAm0/CygGUV7dUZWI6Ibn4yNQ/RZroyUZ/RsaLfpnq18axUoKhMhb1TxjBtPuwO6z7RtLccb9aXPz2Q9I6tr7tHrb8cMeuhAN80+tsBX3IPaEUTzbXg68sg/zJ1bQvpOxW+L+8+E5q8+27Zv9joFDWltwpJ9rMuAT0tUXgAa14LMoP1pLNLR+M4SJT4jd8lj8sqy3b9Md/r6zZEurfDqtVx9SjIgruvENaA9I+/75dMiBynBzPnJOkWd9aNEfk04ogXluD61OEu9jOldmpByywnFszHtNW5vlHvbZ0eLn2hOB0K/VYF3xQnZx4CsXwDeKaE1GdCv/VaOfrdC2oOYLKSP1YmK2MYV6UTsGotol8s1It8XMustJEs2Jf6arr27MlqLMs6XXmC+sLa1TloUCYbENVW23eztet+DSPMmhL1KJEWzVq4Q/6v0V//qdbK6tCJQwzgDmbJ0rvlIKNNAHXXJx9fbIN3XlvJSIcNTJbXxfkamSyXJtilPQnnQVw+Ac5+eJS9cJSnJnS31kV607qQVKU8W9IijRe/aUVZ/qPwvyn2NfOQq7bmOMG/Y6Nhf9+VOeBf51vMqU/ejnf1fqBtdl7WGFANhTlelhriz95ByLFvKmN5Jkm/fhIrSstbThV71fMa0FkfWXv4ztP4S/mq59bMfnV7JKmzSGpAUzJPRiGwJlnr48dos8dIrU9Myz4afWZO6l93yJlmxtG4mMz5rTOWAVs1Lddag6z7jP6I7PKMB3a6tyPFCBH2k7vx8qTej7U95ylLmYyeNijyktHMeP6oDliqfi2vUK5bWGksL38azT9Tt3Wn7zlsUO/QpwpHZwYB2QUQxjW6tz4MkBeT1obJf9u7qUAvupR5ZKKRcfqsQz+76Qr5DKmX8tfIcCdlRs+P0OzsvVB9twzpU/90SckTZbEZ7RyM+pB6hkt+WI1jY8dctnx7QmXeXYhbp1+szUru3mYWg5K9NNif3guEVi/dVND6mOcjF61pue0v54Z6VIQZC3vfFRQSPs/y60Sxj3iZXhs7V3dFbRrOEniSlfF5GjT2yUcunDPIWawza1+vuOkk9E594WAeZV5qdYq8v/GyqshIc2wfQEzVvZt7Vg+cXe3Pk+L3JfVaXPMdIxYGzhec3o9VVnqT8XKeH6QJfo4+pGKucUEfvJpMqYzriY28uUqJmXCTGh0uzUTSRv5nkTWSW9za+lHVvTbmczUsbc0o5CfceJCJcEdQVZ8R0lRlHb4lej7A2NdnCUcLzrkTl4La1xZOfC0u+SLZeqPVIseWNqryFpm57DGPDpYUMyfrFgjsXkb37Qr6PWpVB2bnBx1DwbyBc+ZP2Jj6na4cQReJ79X7fiqi/yX+u6vLEL6AW3MmDhey0q8ZZ7lE/y88t6jZ9Hw7+PCLQNSd9RH6wqeySMi+5Td2f/gvaw6kIWelNz+ZjsLnwI1nm1GQ8EdklfjSR0N9jajoWzcFnJGUu/nzkuT83ihOhv0/XbAyaOj+CMocmPPT9vN+cm97Nedmc6vW1zMWXh7Th+kZC4/BmrDrbMP18LFRqzcg3zwGtw0kNde0t/tdxaD6GU3Nevtwy+q7TVx6zLvuF6swSo9nme8Zw81nN1Rz9eSbF6Eys4+Yno7ag0Uwl1mi+efoYTZo1oHnNhDwp5KWTeHsVGXl9qeDJuUuGRPzHSwaJr5LBl4o+wa+mpHvw1PQ3/Vyj0p/5yGToVMlUpob0ZKbdoT0c0mmBudkZ0HPxtuXxxdUbi986X66016//4fqN++urn26qL6S/I34ufgWB/A3xkfhU3IYAsw0i/EX8XfxD/HPjNxv3Nx5tPJFd335LYX4mSj8bvf8C/bg34w==</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Comparing Distributions for Learning <latexit 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<latexit 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<latexit 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↵✓ <latexit 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ATAC-seq
  5. 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<latexit 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Optimal Transport Entropic
  6. Monge optimal matching: <latexit 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<latexit 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<latexit 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<latexit

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[Monge 1784] <latexit 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Monge’s Problem Points (xi)i, (yj)j Permutation: <latexit 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<latexit 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<latexit 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: {1, . . . , n} ! {1, . . . , n} <latexit 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<latexit 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<latexit 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D(X, Y ) = min n X i=1 d(xi, y (i) )
  7. Monge optimal matching: <latexit 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<latexit 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<latexit 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<latexit

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[Monge 1784] <latexit 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Monge’s Problem Points (xi)i, (yj)j Permutation: <latexit 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<latexit 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<latexit 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: {1, . . . , n} ! {1, . . . , n} <latexit 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<latexit 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D(X, Y ) = min n X i=1 d(xi, y (i) ) ! Seems intractable: n! possibilities. <latexit 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<latexit 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<latexit 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<latexit 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! Di↵erent number of points?
  8. Kantorovitch’s Formulation ↵ = Pn i=1 ai xi = Pm

    j=1 bj yj Points (xi)i, (yj)j Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 xi <latexit 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sha1_base64="lHCrrCdLriSUoJxIcVw6BefNKNo=">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</latexit> Discrete distributions: <latexit 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<latexit 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<latexit 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<latexit sha1_base64="mdlENPIKtv1PnSOEu/s/cH99nYI=">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</latexit>
  9. Kantorovitch’s Formulation ↵ = Pn i=1 ai xi = Pm

    j=1 bj yj Points (xi)i, (yj)j Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 b <latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">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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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<latexit 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<latexit 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Couplings: <latexit 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<latexit 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<latexit 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P j Pi,j = ai <latexit 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<latexit 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<latexit 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<latexit 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P i Pi,j = bj <latexit 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P > 0, P1m = a, P>1n = b <latexit 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<latexit 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<latexit 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sha1_base64="lHCrrCdLriSUoJxIcVw6BefNKNo=">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</latexit> Discrete distributions: <latexit 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<latexit 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<latexit 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<latexit sha1_base64="mdlENPIKtv1PnSOEu/s/cH99nYI=">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</latexit>
  10. Kantorovitch’s Formulation ↵ = Pn i=1 ai xi = Pm

    j=1 bj yj Points (xi)i, (yj)j Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 b <latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">AABB0nictVxfcxPJER8u/w7yD5LHvGxiSHEpQoxzVUnqKlVnbGN8CDBINtydgNJKayFYa4VWsgGdH1J5zVfIa/Ix8jnyDZKnfIX0n5mdWWl2e9YhTNmeHc2vu6d3pqe7Z0Q8SUf5bH39nxc++ta3v/Pd73188dL3f/DDH/348pWfHObZfNpPDvpZmk2fxr08SUfj5GA2mqXJ08k06R3HafIkfr2Fnz85Sab5KBt3Zu8mybPj3nA8Ohr1ezNoenH5ytUuEVnE6Tw5i7rx1ReX19ZvrtO/aLVyS1fWlP63n12JDlRXDVSm+mqujlWixmoG9VT1VA7la3VLrasJtD1TC2ibQm1EnyfqTF0C7Bx6JdCjB62v4fcQnr7WrWN4Rpo5ofvAJYWfKSAjdQ0wGfSbQh25RfT5nChjaxXtBdFE2d7B31jTOobWmXoJrRLO9AzF4Vhm6kj9nsYwgjFNqAVH19dU5qQVlDxyRjUDChNow/oAPp9CvU9Io+eIMDmNHXXbo8//RT2xFZ/7uu9c/ZukvAYlUm09+qyg0FMnRD+itzmHz1ieFDgPgUKix4i1U9L1MY1+DP0X0P4AyhnVjE5iKAtqPatFbkHxIbdE5C4UH3JXRLag+JAtEbkPxYfc10jETknnfnwbig/fFjk/guJDPhKRj6H4kI9F5CEUH/JQRH4FxYf8SkTegeJD3hGR96D4kPdEZAeKD9kRkQdQfMgDEbkDxYfc0cjqlTqFkhGdkbAqN6Fe5oGWIoWWTVG+22QdfdjbAWu6X4GVV/U2/PVjtwN0mlRgdwLm3VEFVp55u2Aj/VjZFt2l3cSHvSti92AG+LF7IvYL9aoC+0XASntdgZXXWgv6+bGy9b0PT37sfRH7AGp+rLxHPYQWP/ZhwI4xqcDui9hH6k0FNsTqTyuwst1vg13xY+V9qgP9/dgQazqvwMr29BA8GD9W3q2eQKsf+0TEPlVvK7BPReyXYN392C8Ddtj3FVizx16iHWRI/kgCK7aOWq9YlVibALWewD8t9paUfOMY2iXMsMAMCXMsInYLxG4golUgWsFy5YUdzcnflbm0C0Q7EBEXexPWZmL/QdEfa2kAYrtAbC8h6jxSfNdmLCfkXZgWCTkrdi6shYwpK+w31hI9H+otr0E8LCF4br+kmX+DoiWMoFBTddReFns8IyN6rkOcUvRmRml4yLhZYRVc1FsRFXtQsYh650G9E1FzD2ouok48qBMRZVe+i+sGzACrf3wXC3riGcA+cnWJwCvYhF3nLqzRCObPPniBj6nlIfxtU+wtlTrJMJrHfRKzHM9KlngKtYVag3YbFW5TfJ3SCktAMu75UMf4+IS5jYVec2yFz4qdPCoyJuF0RiTPsKCD3mJE66kZnXvUckbeHdea4e8W697UmuF3SONn5MVzrRl+pqWfnUP2jsZ2zoFtw2qaaO3belManH9hGqZ+iXZdtLj4Vo/1nEF6bxvS39NvZu8c72WLaqwfW29GI3fGl5fG14SG1XPu6LkZFfSe2Os1tajxSMY67rX1pjJktIuOtRz2qembwT4D/WZMvRmNffC4tijmXjj1prN3UozG1pvROFSc9zwjT97Um9EY0jPrw9ab0cBsS0/H+bbe1LKjBjh2tvWmVn1MWWDMAfGc5xbrFU3JT5praiPyD+qzNa7Pv7qPYc7meREj1FOyvm01nbjYy+olMv5CAlZt1lAO9C/mjg9WprFQG2J8xTLMSvv7Kh27x6PmW6DFCFY/nwFIOfMUJDQ5CbTeKVC8JUZd5ZEZ3IaIw1lytITq6taZ6C1avpw1Kre9oFYpLrOjtXrskr3Oae5NyCdskWYlPbQq33AVRUlDrZKGZHpNdPder9ey9tdF3GQJMSlmWp9OhPgkrT5O9Wm97ej4mj7lmUHhMx87fzHbfKStDcY8GdkilKWOp9vP5JHcNtxXbyib4+bPInqjaK9OyGqM6EQqF6NQky1mb3xBz5b2AZ3JIQ+m0Yf3GGkqE8WnZphFx3x6RBbVtbcSb9SXydBxPSera+xxPXrooIcedPMYZwt2jAdQ60DMcABPnYAo51Khq4w0PlW/Lk5HM3qD9RF9WrKQhgbbm6RkIeui7JclKqeAxtnAUXo4jWU6Bt9doSRH/T55bOxatvzX6OTWnG/3aI5Xz+bqTMyAuG4Q14hWDZ/q8tMyB5Zg4f1kg/zX+lEivyYc0YZKXJ87nFkvYzrxTyiCnZBnnNJqk1ZHubebn1r+xHDaV+bsHE+zM7KQEdm/CPanjOZkRD/u3QFzgs4WISUbGWJ3RoV34/N1RuIcs37cSPGtBjvfErJlc+Jv6LqrK6e5yBED7wNnS3Pb6KRFvmBCXKfautu1Xb/7INLek3BnCVO0c+U68f+EfpsfM0/WVmYEahjfQK5tne99ZBSzoI56tMvX2yDT15XyaiHDcy213f+sTFdLkm1TxIXy4G49AM59emZeOEumJHe+0of30bpsLlKeLOkRR3tEUTzb/aHegVHuG7RLrtGa69IsGcIsmBVRhOkrZZGX+dbzKlMPo53/X6hbXZe1hhQjZTO4rCEpv59QtOZKmcKs5vn7mlaTX+vTpV71fMY0F4+dtfwNtP4cfhu5zXMYnbhkFW7THGAK9slqhFuilR5hvG6XeJmZaWjZZ8vPzknTy205T3zN1s3G2CeNqezTrHmrsxamfh4arxwarwJ12KGzRqtF024s0Qsxtujo08pQfk24dRpQnouUZY/MoEYBUrqxVBjVgUhVjvEN6r1Ia12k1YPV6p4GuGs+BOlf68ur+5tid4/UHfJt+uSBcfwyoFU6Ip/LtNZHakwBOX+q7au7+rvUgtxjsqBIme9x4orhU6c+lbNC0l/qnS0jO28tgrm3dKr7GBvbpfpvV5DHtCZyWpcG8Sn1SLT8rhzRkkW66fgcEWX+e+RTsd9RHzO7ve07iUr+hI03eVVZXhwpjEn/UuZtbyV63XPi14hiwrn2rmOg1fwNIwXGmEyC37PM6Q3hLscnCezRxmQ/V+0Un+KNHYluktQL9ccAG8NRr53r7twyIzZj+xX0RK3bt+7rIfNLgzlK/M5zotejXe1Y+6iLpefz0erpXa78XKeH+RJfq4859XEjCxvllTFd9VkwF5aoGRfGhHBpNoom8jeTvInMfDoVStn0NpTLmQa2MS8pXpLugSLC591d93pznwjjiFfoxYR1qXGLRAmzcZnOD7iWFrNSF1f2IW69WLsbpc5OVLVTGOrubmHtN1vIhKxfqqScDfd2Ze+WohQ5C8MU+opv9FbFhy7Nz6Dg70j5okPDMSR32Ab/dlNtqZ0PcBvija5zRjOiFrQFg6XYu6fHWe5Rr6M3DnWXfgiHcB4j0LUk/Yh20qayM2VZcpd6OP1TsgJTlYjS257Nx+BykUeyyqnJeEZk2eTRjJT5Lk7TsRgOISMpcwnnw+ca0iiOlPlOU7MxGOryCMocmvAw9xjC3rnt3ZyXy6leX6tcQnnwLmBOXAwOT/6qYxXbL8RCTZ038uE5oHU4qqFudov/dRyGj+XUnFcot5y+a/Yq4K1zv0RnZNEfbr5mLLeQ2VzNMZxnVozOekt+fuz3RY3eVOaM5sPTR3/UzgHDa6E4DypLx3h3Fll5Q6nguYBPhkz9R/3jgvxthDcFjSo5mlAy5xTV1EwPmZr5xqVvdOazEJksnSqZytRsHNGmG7Fbak/dgZ+twgNsejuUv0vJfxHr//7sAFqPyHqYLDpnDrrUllD2w56iDejZ3p+tkhjv8vLd3g604Fl4i1rxnu8D6o93fTulsVV/g4TX+n2VqUEpIlk+3bPrKoYRlE/eOAdkvucb0V16zmLxzbPjgLNFc39qWaIFfSLfLIgr8bEjZZ/m6kSf1ePJAd6w7xX5oUj9htp62s7jnitx3q/kvL/EOSftlDm8dT6rv5tVxWXL4TIocmcnul9GcbY9z6vPjW5XcuE76PX4YQ1+6EjZJu2/pkh4quqzefMamnMtk3vCOlYmE8l6wDizV7zv+sj2pIbXScD471Wi7zmS7oIsMeW/IzphmxK9VOtmh6Tnm471mdS7NdLq71G+uLx2a/n/MlitHGzc/MPNW4821j6/rf+bg4/Vz9Qv1HVY4r9TnwOxfXUADE7VX9Xf1N83DzYXm3/a/DN3/eiCxvxUlf5t/uW/FJCnCA==</latexit> <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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Couplings: <latexit 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<latexit 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<latexit 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P j Pi,j = ai <latexit 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<latexit 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<latexit 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<latexit 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P i Pi,j = bj <latexit 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