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Optimal Transport for Imaging and Learning

Optimal Transport for Imaging and Learning

Talk @UCL, Louvain la Neuve, updated 30 Sept. 2018, talk @Lille

Gabriel Peyré

August 30, 2017
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  1. Optimal Transport for Imaging and 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 RESEARCH UNIVERSITY PARIS Joint works with: Francis Bach François-Xavier Vialard Jean Feydy Thibault Séjourné Lénaic Chizat Aude Genevay Marco Cuturi Alain Trouvé Shun'ichi Amari
  2. Source image (X) Style image (Y) Sliced Wasserstein projection of

    X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization ! images, vision, graphics and machine learning, . . . Probability Distributions in Data Sciences Probability distributions and histograms <latexit 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  3. Source image (X) Style image (Y) Sliced Wasserstein projection of

    X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization ! images, vision, graphics and machine learning, . . . Probability Distributions in Data Sciences Probability distributions and histograms <latexit 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<latexit sha1_base64="NHBrzuHY/DIiTVSeYz9st2EbBcI=">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</latexit> Parametric model: ✓ 7! ↵✓ <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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✓ ↵✓ <latexit 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  4. Source image (X) Style image (Y) Sliced Wasserstein projection of

    X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization ! images, vision, graphics and machine learning, . . . Probability Distributions in Data Sciences Probability distributions and histograms <latexit 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min ✓ D(↵✓, ) <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=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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✓ ↵✓ <latexit 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  5. 3. Sinkhorn Divergences <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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x <latexit 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<latexit 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d(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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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  6. Kantorovitch’s Formulation Points (xi)i, (yj)j Input distributions ↵ = Pn

    i=1 ai xi = Pm j=1 bj yj Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 xi yj
  7. Kantorovitch’s Formulation Points (xi)i, (yj)j Input distributions ↵ = Pn

    i=1 ai xi = Pm j=1 bj yj 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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xi yj Couplings: <latexit 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<latexit 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<latexit 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U(a, b) def. = n P 2 Rn⇥m + ; P1n = a, P>1m = b o <latexit 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<latexit 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<latexit 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  8. Kantorovitch’s Formulation Points (xi)i, (yj)j Input distributions ↵ = Pn

    i=1 ai xi = Pm j=1 bj yj 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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[Kantorovich 1942] min nP i,j d(xi, yj)pPi,j ; P 2 U(a, b) o <latexit 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<latexit 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xi yj d(xi ,yj ) <latexit 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Couplings: <latexit 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<latexit 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U(a, b) def. = n P 2 Rn⇥m + ; P1n = a, P>1m = b o <latexit 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<latexit 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  9. Optimal Transport Distances b <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="ty3Ki+Ibgkag0xq97PVpv2d8vdk=">AABB03ictVzdchPJFW42P7uQP3ZzmZtJDCk2xRLj3aoktZWqNbYxXgQIJBt2V0DpZyyExxqhkYxB+CaV27xCbpO3yHPkDZKrvELOT/d0j9Qzp8chdNnuafV3zukz3afPOd2iN0lG2Wx9/Z8XPvje93/www8/unjpRz/+yU9/dvnjTw6ydD7tx/v9NEmnT3rdLE5G43h/Npol8ZPJNO4e95L4ce9oCz9/fBJPs1E6bs/eTOKnx93heHQ46ndn0PT88idXOkRk0Uu6/aOzqNO88vzy2vqNdfoXrVZu6sqa0v+a6cfRvuqogUpVX83VsYrVWM2gnqiuyqB8p26qdTWBtqdqAW1TqI3o81idqUuAnUOvGHp0ofUIfg/h6TvdOoZnpJkRug9cEviZAjJSVwGTQr8p1JFbRJ/PiTK2ltFeEE2U7Q387Wlax9A6Uy+gVcKZnqE4HMtMHarf0xhGMKYJteDo+prKnLSCkkfOqGZAYQJtWB/A51Oo9wlp9BwRJqOxo2679Pm/qCe24nNf952rf5OUV6FEqqVHn+YUuuqE6Ef0NufwGcuTAOchUIj1GLH2mnR9TKMfQ/8FtN+HckY1o5MelAW1nlUit6D4kFsicheKD7krIhtQfMiGiGxC8SGbGonYKencj29B8eFbIueHUHzIhyLyERQf8pGIPIDiQx6IyG+h+JDfisjbUHzI2yLyLhQf8q6IbEPxIdsich+KD7kvIneg+JA7Glm+UqdQUqIzElblJtSLPNBSJNCyKcp3i6yjD3srYE33S7Dyqt6Gv37sdoBO4xLsTsC8OyzByjNvF2ykHyvboju0m/iwd0TsHswAP3ZPxH6tXpZgvw5YaUclWHmtNaCfHytb33vw5MfeE7H3oebHynvUA2jxYx8E7BiTEmxTxD5Ur0qwIVZ/WoKV7X4L7IofK+9Tbejvx4ZY03kJVranB+DB+LHybvUYWv3YxyL2iTotwT4Rsd+AdfdjvwnYYd+WYM0ee4l2kCH5IzGs2Cpq3XxVYm0C1LoC/yTfWxLyjXvQLmGGOWZImGMRsZsjdgMRjRzRCJYry+1oRv6uzKWVI1qBiF6+N2FtJvYf5P2xlgQgtnPE9hKiyiPFd23GckLehWmRkLN858JayJjS3H5jLdbzodryGsSDAoLn9gua+dcpWsIICjVVRe1FvsczMqLnKsRrit7MKA0PGTfLrYKLOhVRPQ+qJ6LeeFBvRNTcg5qLqBMP6kRE2ZXv4joBM8DqH9/Fgp54BrCPXF4i8Ao2Yde5A2s0gvnTBC/wEbU8gL8tir2lUiUZRvO4T2KW42nBEk+htlBr0G6jwm2KrxNaYTFIxj0f6BgfnzC3sdBrjq3wWb6TR3nGJJzOiOQZ5nTQW4xoPdWjc5dazsi741o9/J183ZtaPfwOafyMvHiu1cPPtPSzc8je1tj2ObAtWE0TrX1br0uD8y9Mw9Qv0a6LFhff6rGeM0jvtCb9Pf1m9s7xXraoxvqx9Xo0Mmd8WWF8dWhYPWeOnutRQe+JvV5Ti2qPZKzjXluvK0NKu+hYy2Gf6r4Z7DPQb8bU69Fogse1RTH3wqnXnb2TfDS2Xo/GgeK85xl58qZej8aQnlkftl6PBmZbujrOt/W6lh01wLGzrde16mPKAmMOiOc8t1ivaEp+0lxTG5F/UJ2tcX3+1X0MczbP8hihmpL1bcvp9PK9rFoi4y/EYNVmNeVA/2Lu+GBFGgu1IcZXLMOssL+v0rF7PGq+AVqMYPXzGYCUM09AQpOTQOudAMWbYtRVHJnBbYg4nCWHS6iObp2J3qLly1mjYttzapXiMjtaq8cO2euM5t6EfMIGaVbSQ6P0DZdRlDTUKGhIpldHd2/1ei1qf13ETZYQk3ym9elEiE/SquNUn9Zbjo6v6lOeGRQ+87HzF7PNh9raYMyTki1CWap4uv1MHsltw331urI5bv4sojeK9uqErMaITqQyMQo12WL2xhf0bGnv05kc8mAafXiPkaYyUXxqhll0zKdHZFFdeyvxRn2ZDB3XM7K6xh5Xo4cOeuhB149xtmDHuA+1NsQM+/DUDohyLuW6SknjU/VZfjqa0husjuiTgoU0NNjexAULWRVlvyhQeQ1onA0cpYfTWKZj8J0VSnLU75PHxq5Fy3+VTm7N+XaX5nj5bC7PxAyI6wZxjWjV8KkuPy1zYAkW3k82yH+tHiXyq8MRbajE9ZnDmfUyphP/mCLYCXnGCa02aXUUe7v5qeVPDKemMmfneJqdkoWMyP5FsD+lNCcj+nHvDpgTdLYICdnIELszyr0bn68zEueY9eNGim812PkWky2bE39D111dGc1Fjhh4HzhbmttGJw3yBWPiOtXW3a7t6t0HkfaehDtLmKKdK9eI/6f02/yYebK2MiNQw/gGMm3rfO8jpZgFddSlXb7aBpm+rpRXchmeaant/mdlulKQbJsiLpQHd+sBcO7TM/PCWTIlubOVPryPVmVzkfJkSY842kOK4tnuD/UOjHJfp11yjdZch2bJEGbBLI8iTF8pi7zMt5pXkXoY7ez/Qt3quqg1pBgpm8FlDUn5/ZiiNVfKBGY1z98jWk1+rU+XelXzGdNcPHbW8jto/SX8NnKb5zA6vYJVuEVzgCnYJ6sRbolWeoTxulXgZWamoWWfLT87J00vt+U88TVbNxtjn9Sm0qRZc6qzFqZ+HhovHRovA3XYprNGq0XTbizRczG2aOvTylB+dbi1a1Cei5Rlj8ygRgFSurFUGNWBSFWO8Q3qrUhrXaTVhdXqnga4az4E6V/ry6v7Xb67R+o2+TZ98sA4fhnQKh2Rz2VaqyM1poCcv9D21V39HWpB7j2yoEiZ73HiiuFTpz6Vs1zSX+udLSU7by2Cubf0WvcxNrZD9c9XkMe0JjJalwbxBfWItfyuHNGSRbrh+BwRZf675FOx31EdM7u97TuJCv6EjTd5VVleHCmMSf9S5m1vJXrdc+LXiGLCufaue0Cr/htGCowxmQS/Z5nRG8Jdjk8S2KPtkf1ctVN8ijd2JLpBUi/UHwNsDEe9dq67c8uM2IztN9ATtW7fuq+HzC8J5ijxO8+JXpd2tWPtoy6Wns9Hq6t3ueJzlR7mS3ytPubUx40sbJRXxHTUl8FcWKJ6XBgTwqXeKOrIX0/yOjLz6VQoZdPbUC5mGtjGvKB4SboHigifd3fN6819Koyjt0KvR1iXGrdIlDAbl+r8gGtpMSt1cWUf4taLlbtR4uxEZTuFoe7uFtZ+s4WMyfolSsrZcG9X9k4hSpGzMEyhr/hGb1l86NL8Egr+jpQvOjQcQ3KHLfBvN9WW2nkPtyFe6TpnNCNqQVswWIq9u3qcxR7VOnrlUHfph3AI5zECXUvSj2gnrSs7U5Yld6mH039NVmCqYlF627P+GFwu8khWOdUZz4gsmzyakTLfxak7FsMhZCRFLuF8+FxDGsWhMt9pqjcGQ10eQZFDHR7mHkPYO7e96/NyOVXra5VLKA/eBcyJi8HhyV95rGL7hVioqfNG3j8HtA6HFdTNbvG/jsPwsZzq8wrlltF3zV4GvHXuF+uMLPrD9deM5RYym8s5hvNM89FZb8nPj/2+qNabSp3RvH/66I/aOWB4LRTnQWXpGO/OIitvKBU8F/DJkKr/qH9ckL+N8CqnUSZHHUrmnKKcmukhUzPfuPSNznwWIpOlUyZTkZqNI1p0I3ZL7anb8LOVe4B1b4fydyn5L2L9358dQOshWQ+TRefMQYfaYsp+2FO0AT3b+7NlEuNdXr7b24YWPAtvUCve871P/fGub7swtvJvkPBav6dSNShEJMune3Zd9WAExZM3zgGZ7/lGdJees1h88+w44GzR3J9almhBn8g3C3ql+J4jZZ/m6kSf1ePJAd6w7+b5oUj9ltq62s7jnitxbpZybi5xzkg7RQ6nzmfVd7PKuGw5XAZ57uxE90spzrbnedW50e1SLnwHvRo/rMAPHSlbpP0jioSnqjqbN6+gOdcyuSesY2UykawHjDO7+fuujmxPKnidBIz/bin6riPpLsjSo/x3RCdsU6KXaN3skPR807E6k3qnQlr9Pcrnl9duLv9fBquV/Y0bf7hx8+HG2le39H9z8JH6hfqVugZL/HfqKyDWVPvA4FT9Vf1N/X3zYPPd5p82/8xdP7igMT9XhX+bf/kvm82nVQ==</latexit> ⇡ = P i,j Pi,j xi,yj <latexit 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<latexit 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<latexit 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⇡ ↵ Wp(↵, )p def. = min ⇡2M1 + (X2) ⇢Z X2 d(x, y)pd⇡(x, y) ; ⇡1 = ↵, ⇡2 = <latexit 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<latexit 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  10. Optimal Transport Distances b <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="ty3Ki+Ibgkag0xq97PVpv2d8vdk=">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</latexit> ⇡ = P i,j Pi,j xi,yj <latexit 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<latexit 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<latexit 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⇡ ↵ Wp(↵, )p def. = min ⇡2M1 + (X2) ⇢Z X2 d(x, y)pd⇡(x, y) ; ⇡1 = ↵, ⇡2 = <latexit 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<latexit 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<latexit sha1_base64="sjDT6hvhNW1OHhd1E52kbdPXLFU=">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</latexit> Theorem: <latexit 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<latexit 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<latexit 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Wp is a distance and <latexit 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↵n * , Wp(↵n, ) ! 0 <latexit 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Weak⇤ (aka in law) convergence: <latexit 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<latexit 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. . . x1 <latexit 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x2 <latexit 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<latexit 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<latexit 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x3 <latexit 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R X fd↵n ! R X fd <latexit 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<latexit 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<latexit 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|| xn x ||1 = 2 <latexit 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<latexit 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<latexit 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Wp( xn x) = |xn x| <latexit 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<latexit 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<latexit 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<latexit 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  11. Algorithms Linear programming: O(n 3 log(n)2) Hungarian/Auction: O(n 3) =

    1 n Pn j=1 yj ↵ = 1 n Pn i=1 xi ↵ ↵ ↵ 1-D case: sorting O(n log(n)).
  12. Algorithms Linear programming: O(n 3 log(n)2) Hungarian/Auction: O(n 3) =

    1 n Pn j=1 yj ↵ = 1 n Pn i=1 xi ↵ ↵ ↵ 1-D case: sorting O(n log(n)). W1(↵, ) = min div(v)=↵ R ||u(x)||dx p = 1 d = || · || ! min-cost flow, on graphs O(n2 log(n)).
  13. Monge-Amp` ere/Benamou-Brenier, d = || · ||2 2 . Algorithms

    Linear programming: O(n 3 log(n)2) Hungarian/Auction: O(n 3) = 1 n Pn j=1 yj ↵ = 1 n Pn i=1 xi ↵ ↵ ↵ 1-D case: sorting O(n log(n)). W1(↵, ) = min div(v)=↵ R ||u(x)||dx p = 1 d = || · || ! min-cost flow, on graphs O(n2 log(n)).
  14. Monge-Amp` ere/Benamou-Brenier, d = || · ||2 2 . Semi-discrete:

    Laguerre cells, d = || · ||2 2 . Algorithms Linear programming: O(n 3 log(n)2) Hungarian/Auction: O(n 3) = 1 n Pn j=1 yj ↵ = 1 n Pn i=1 xi ↵ ↵ ↵ 1-D case: sorting O(n log(n)). W1(↵, ) = min div(v)=↵ R ||u(x)||dx p = 1 d = || · || ! min-cost flow, on graphs O(n2 log(n)).
  15. Need for fast approximate algorithms for generic c. Monge-Amp` ere/Benamou-Brenier,

    d = || · ||2 2 . Semi-discrete: Laguerre cells, d = || · ||2 2 . Algorithms Linear programming: O(n 3 log(n)2) Hungarian/Auction: O(n 3) = 1 n Pn j=1 yj ↵ = 1 n Pn i=1 xi ↵ ↵ ↵ 1-D case: sorting O(n log(n)). W1(↵, ) = min div(v)=↵ R ||u(x)||dx p = 1 d = || · || ! min-cost flow, on graphs O(n2 log(n)).
  16. 3. Sinkhorn Divergences <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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x <latexit 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<latexit 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d(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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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  17. Entropic Regularization Relative-entropy: <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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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>
  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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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Ki,j def. = e d(xi,yj ) " <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 sha1_base64="9DTYcVgVzHkjGQd3IVuH7Cafdno=">AABElnictVz9chu3EYfSr1hNW6eddjrTfy6V3XE6riupmWmnmXRii7KsSLFpk5KdhLaGRx7pk088mkfSHwynfbM+R1+g0/7VV+juAjjgSNwtTnWNkYQD8dtd7AGL3QXocJzE2XR7+x8b733nu9/7/g/ev7L5ww9+9OOfXP3wp6dZOpv0opNemqSTJ2E3i5J4FJ1M42kSPRlPou5FmESPwxd7+PnjeTTJ4nTUnr4ZR08vusNRPIh73Sk0nV3927VO82yx6BClxSTqL+PlTf0YJrNoeb5cBpufBZt2n6Aza5zFAfz982bniMVvQrdCW9CZN87OAb957ezq1vatbfoXrFd2VGVLqH/N9MOP/io6oi9S0RMzcSEiMRJTqCeiKzIo34gdsS3G0PZULKBtArWYPo/EUmwCdga9IujRhdYX8HsIT9+o1hE8I82M0D3gksDPBJCBuA6YFPpNoI7cAvp8RpSxtYz2gmiibG/gb6hoXUDrVDyHVg6ne/ricCxTMRB/ojHEMKYxteDoeorKjLSCkgfWqKZAYQxtWO/D5xOo9wip9RwQJqOxo2679Pm/qCe24nNP9Z2Jf5OU16EEoqVGn+YUumJO9AN6mzP4TMqTAOchUIjUGLH2inR9QaMfQf8FtN+HsqSa1kkIZUGty0rkHhQXco9FHkBxIQ9Y5DEUF/KYRTahuJBNhUTshHTuxreguPAtlvNDKC7kQxb5CIoL+YhFnkJxIU9Z5NdQXMivWeRdKC7kXRZ5BMWFPGKRbSguZJtFnkBxIU9Y5D4UF3JfIctX6gRKSnRiZlXehnqRB1qKBFpus/LdIevowt7xWNO9Eiy/qhvw141teOg0KsHue8y7QQmWn3kHYCPdWN4W3aPdxIW9x2IPYQa4sYcs9gtxXoL9wmOlvSjB8mvtGPq5sbz1/RKe3NgvWex9qLmx/B71AFrc2AceO8a4BNtksQ/FyxKsj9WflGB5u98Cu+LG8vtUG/q7sT7WdFaC5e3pKXgwbiy/Wz2GVjf2MYt9Il6XYJ+w2K/AuruxX3nssG9LsHqP3aQdZEj+SAQrtopaN1+VWBsDtS7DP8n3loR84xDaOcwwxwwJc8EiDnLEgSfiOEcce8uV5XY0I3+X59LKES1PRJjvTVibsv37eX+sJR6IRo5orCCqPFJ813osc/IudAuHnOY7F9Z8xpTm9htrkZoP1ZZXIx4UEHJuP6eZf5OiJYygUFNV1J7ne7xEBvRchXhF0ZsepebB46a5VbBRr1lU6ECFLOqNA/WGRc0cqBmLmjtQcxZlVr6N63jMAKN/fBcLepIzQPrI5SUAr+A27Dr3YI0GMH+a4AU+opYH8LdFsTdXqiTDaB73ScxyPC1Y4gnUFmIL2k1U2KD4OqEVFoFksucDFePjE+Y2FmrNSSu8zHfyIM+Y+NOJSZ5hTge9xYDWUz06R9SyJO9O1urh7+XrXtfq4fdJ40vy4mWtHn6qpJ9eQva2wrYvgW3Bahor7Zt6XRoy/yJp6Pom7bpocfGtXqg5g/Re16R/qN7M4SXeyx7VpH5MvR6NzBpfVhhfHRpGz5ml53pU0HuSXq+uBbVHMlJxr6nXlSGlXXSk5DBPdd8M9umrN6Pr9Wg0wePao5h7YdXrzt5xPhpTr0fjVMi855I8eV2vR2NIz1Ifpl6PBmZbuirON/W6lh01IGNnU69r1UeUBcYckJzzssV4RRPyk2aKWkz+QXW2xvb51/cxzNk8y2OEakrGty2nE+Z7WbVE2l+IwKpNa8qB/sXM8sGKNBZil42vpAzTwv6+Tsfs8aj5Y9BiAKtfngFwOfMEJNQ5CbTeCVDcYaOu4sg0bpfF4SwZrKA6qnXKeouGr8waFdvOqJWLy8xojR47ZK8zmntj8gmPSbOcHo5L33AZRU5DxwUN8fTq6O6tWq9F7W+zuPEKYpzPtB6dCMmTtOo41aX1lqXj6+qUZwpFnvmY+YvZ5oGyNhjzpGSLUJYqnnY/nUey23BfvSlMjlt+FtAbRXs1J6sR04lUxkahOlssvfEFPRvaJ3QmhzwkjR68x0BRGQt5aoZZdMynB2RRbXvL8UZ96QydrGdkdbU9rkYPLfTQga4f4+zBjnEfam2IGU7gqe0R5WzmukpJ4xPxu/x0NKU3WB3RJwULqWlIexMVLGRVlP28QOUVoHE2yCjdn8YqHY3vrFHio36XPCZ2LVr+63Ryq8+3uzTHy2dzeSamT1x3iWtAq0ae6sqnVQ5SgoXzk13yX6tHifzqcEQbynF9ZnGWehnRiX9EEeyYPOOEVhu3Ooq97fzU6ieaU1Pos3M8zU7JQgZk/wLYn1KakwH92HcH9Am6tAgJ2UgfuxPn3o3L14nZOWb8uFjIWw1mvkVky2bEX9O1V1dGc1FGDHIfWK7Mba2TY/IFI+I6UdbdrO3q3QeR5p6EPUskRTNXbhD/j+m3/tHzZGttRqCG8Q1kyta53kdKMQvqqEu7fLUN0n1tKa/lMjxTUpv9z8h0rSBZgyIulAd36z5w7tGz5IWzZEJyZ2t95D5alc1FyuMVPeJoBxTFS7s/VDswyn2TdsktWnMdmiVDmAXTPIrQfbks8irfal5F6n60s/8LdaProtaQYiBMBldqiMvvRxSt2VImMKvl/H1Bq8mt9clKr2o+I5qLF9Za/hZaP4LfWm797EcnLFiFOzQHJAXzZDQiW4K1Hn687hR46ZmpaZlnw8/MSd3LbrlMfC2tm4mx57WpNGnWvFZZC12/DI1zi8a5pw7bdNZotKjbtSU6Y2OLtjqt9OVXh1u7BuUZS5n3yDQq9pDSjqX8qPZZqnyMr1FvWVrbLK0urFb7NMBe8z5I91pfXd3f5rt7IO6Sb9MjD0zGL31apTH5XLq1OlKTFJDzJ8q+2qu/Qy3IPSQLipTlPU5cMfLUqUdlmUv6G7WzpWTnjUXQ95ZeqT7axnao/oc15AWtiYzWpUZ8Qj0iJb8tR7BikW5ZPkdAmf8u+VTS76iOme3e5p0EBX/CxJtyVRleMlIYkf65zNvhWvR6aMWvAcWEM+Vdh0Cr/htGChKjMwluzzKjN4S7nDxJkB5tSPZz3U7JU7yRJdEtknohPvOwMTLqNXPdnlt6xHpsv4WeqHXz1l09eH6JN0eO32VO9Lq0q10oH3Wx8nw5Wl21yxWfq/QwW+Fr9DGjPnZkYaK8IqYjPvXmIiWqx0VifLjUG0Ud+etJXkdmeTrlS1n31pSLmQZpY55TvMTdA0WEy7u74fTmPmbGEa7RCwlrU5MtHCXMxqUqP2BbWsxKXVnbh2TrlcrdKLF2orKdQlO3dwtjv6WFjMj6JYLL2cjetuydQpTCZ2EkhZ6QN3rL4kOb5qdQ8HcgXNGh5uiTO2yBf3tb7In9d3Ab4qWqy4xmQC1oC/orsXdXjbPYo1pHLy3qNn0fDv48YtA1J31MO2ld2SVlXnKbuj/9V2QFJiJipTc964/B5sKPZJ1TnfHEZNn40cRCfxen7lg0B5+RFLn485HnGtwoBkJ/p6neGDR1fgRFDnV46HsMfu/c9K7Py+ZUra91Lr485C6gT1w0Dk/+ymMV08/HQk2sN/LuOaB1GFRQ17vF/zoOzcdwqs/Ll1tG3zU793jrsl+kMrLoD9dfM4abz2wu5+jPM81HZ7wlNz/p9wW13lRqjebd00d/1MwBzWshZB6Ul07i7Vlk5PWlgucCLhlS8R/x9w3+2wgvcxplctShpM8pyqnpHjw1/Y1L1+j0Zz4yGTplMhWpmTiiRTdi98ShuAs/e7kHWPd2qPwupfyLWPf3Z/vQOiDrobPoMnPQobaIsh/mFK1Pz+b+bJnEeJdX3u1tQwuehR9TK97zvU/98a5vuzC28m+QyLX+pUhFvxCRrJ7umXUVwgiKJ28yB6S/5xvQXXqZxZI3zy48zhb1/alViRb0CX+zICzFh5aUPZqrY3VWjycHeMO+m+eHAvF7ausqO497Lse5Wcq5ucI5I+0UOby2Pqu+m1XGZc/i0s9zZ3PVL6U425znVedGG6Vc5B30avywAj+0pGyR9l9QJDwR1dm8WQXNmZLJPmEdCZ2JlHrAOLObv+/qyHZewWvuMf6jUvSRJekByBJS/jugE7YJ0UuUbvZJennTsTqTeq9CWv09SknT3HU080DfWqzO0idq3snMhf7fCBa0HnA31TcTuexJVEInpBuBEVGSdyS5cwNeHl4aH1k4KjNWkpnHt3fnHrLMPegMGGkGLIUhK4lawWdXt3ZW/3+M9crp7q2d7Vs7D3e3Pr+j/u+M98WvxK/FDdg3/ig+hxnaFCfA6Z8bH2z8fOMXjV82/tLYbxzIru9tKMzPROFfo/lfyd1HHA==</latexit>
  19. Sinkhorn’s Algorithm Proposition: <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 sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit 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<latexit

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<latexit 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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Ki,j def. = e d(xi,yj ) " <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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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>
  20. Sinkhorn’s Algorithm Proposition: <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 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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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Ki,j def. = e d(xi,yj ) " <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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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> 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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<latexit 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<latexit sha1_base64="c65tFjuRWb0VgfFjoHmUj+uQL6Q=">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</latexit>
  21. 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 sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> Only matrix/vector multiplications. K <latexit 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<latexit 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Matrix-vectors <latexit 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<latexit 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parallelization <latexit 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<latexit 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GPU <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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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Ki,j def. = e d(xi,yj ) " <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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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> 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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<latexit 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  22. [Solomon et al, SIGGRAPH 2015] Generalizations ↵1 <latexit sha1_base64="T+Pmv5LDOkLawyFmur1rzuD2eMk=">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</latexit> <latexit

    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<latexit 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↵2 <latexit 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<latexit 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min P k kWp p (↵k, ) <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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  23. [Liereo, Mielke, Savar´ e 2015] [Chizat, Schmitzer, Peyr´ e, Vialard

    2015] Balanced OT Unbalanced OT min ⇡ R cd⇡+ <latexit 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<latexit 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<latexit 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<latexit 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⇢KL(⇡1 |↵) + ⇢KL(⇡2 | ) <latexit 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<latexit 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<latexit 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<latexit 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[Solomon et al, SIGGRAPH 2015] Generalizations ↵1 <latexit 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<latexit 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↵2 <latexit 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<latexit 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min P k kWp p (↵k, ) <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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  24. ↵t+⌧ = min ↵ Wp p (↵t, ↵) + ⌧f(↵)

    <latexit 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qB1LrypYJ1yyCo/zwG+hH+P4K9Lv1xTRzoPz6KeEUWffl2UQk+8nty6cqW5UsJR3X21ox6oQ2jj2+ptozqel54q2aWfKvvGK64lkzJXHxA+5L9GpKNYmqsWTR/VPsT3+bvOXoI8WOtLVi3ZcwfF3GdkK+wDJX+1rFVU24rnZuQUQrEOOY6v/6BUPIfE8+48DsbdmnxSJX8z3CQnn8yEFGWzS5rX+i1oO6x3V7tuEyfpN+wVva6aWtZeLkEeactobsaY08pLEW1omJzFuqq2Vpc2ijfj1bUw225lpaqurX1fLO233c6q5F92yP9tWhhHVvV82Y0KC/ud7fx/287beKgbb2VBV/VVv7Ol31Rbuqofsu3IvwujSzXMN3q5zptMCrfMtfQtc3L3VmgcmOcge95TK2ZqWKmy4m9WyPL4fCzi2+N6NX4muqp7J3PMGJxPHMhdm+ZO3L5egeL4rU3zDGLPc4rnptQ6+A7elpJTsB8quRu5Gjkqoe7kouMipkXcpqTRIvKDbDyZv48U90XjDIJ9ekNmDxN9Egtvp0NLTXLI2xk9sa8Q3SI1vr1bZu9GBQ3z6Q5zg3CSK18+r+y15egxySQyMsrO6YtMusRTbt4XJVIJ9Tx/vhu5STmR+q2Mvi2bkaBDK5ZVukrIqn3aNCde1hXvPe9RHfMJlJnCXbUzB8dqTm/PoUk64Xks24/KvEFRwyZ/2fMyRfajQ/IMMRRN/jLFqlmkYrmxfaxr75xUotzatLHVHKvqdGHFzyHOMTREi246odpCSmtRlEK1lKfk8saJrkeUpUutvE1Wyfv2pjRmmGg/yzuGULvhe9QXlfle5nwi3/U/ID3iboiWLnMVj2LO6nMefL5iSijzbs7gSRvbUse0czfRs1E413vhKWWfZtumqqvXcUSze+q+eqD+Gmh1IffX3lJMdR7BJqX9WP79FAfZbX72DuHw7Ugb6ml2R2PVPkJ551HKU33Pm5/uToBqme5ORtdPOSSvT2I/5fvqMY0b6miDpT6O2FH8gO5e81HGm+aOKVejdEN+Nd0tkOF+qe4lNbQjSfZK8U3QIo19G+A1OkHI9y6gHU5rSdeitDguJudVOO3rmVvX+S3zXUiXT7QuO/oWIF/uC70rMrQX8oJKwTlfKdmXOw7Snyg5jylj7As6my0jahMvhKnkzwyZk5RcBpumjNtDdC8y+Xw2zXn4zrO5Kt4n5rL0GJ5yJxLrI0+1qOEYars5emy3vDe7KHE8daa9n5Xc0M5TXMm4zalccfW5r0YW3RVAXo+0J/FvRRuI4crniuzPK5auLum2rirdheWT3wwa0u2JxlqrbAtvX3S1hjpew/BclGSYZ3ZWtNo4ivecNG19Lby2Ft5NWi2z9Ec+CofQ1/CNx4cRfRfnNP0U3iWabw3hsYZQ2aAzFy8dNI6CVO5lVNACuppSns5RROl5bJo/d+Q+k+0/rdrK7lrF20+PqQZu1q7NfSW/1WA+4fc+en6KvHObx7TS5jvWKd81HSFxfGNOSeH5j1uU9gIettJjfRY8Ue+XJLHRMxp5vFI3VJL5FH8d8J1Q0pPaaSsWJRxfXY+yL9u6HkVak7Elf97NLO8TFXfDISIEY982kKcQL+0ejOrc9Fj6eEqHmk6o5e9l/PIjqhFxexRAf66q7pr9XC2cvaFYDM8XyS8A2b/OMyKEOc3CvPP58ReDMKUPljyg+Hem6v16E1LgnTyI2qa3rt71w75hSKNnjNDkTgjz+0N8+hxbGs5a+c+m4anzoXLNEhd/i4TnffhsDkrJ58xeQpmGdNb9I+2xjiK8w4AoFG/HsTm0aI6Jdxbg2Pm++vsg1dekm2bhZFZI/kcZKimUIXQGAr01nzUI8bNL8xLapY0tc5Xbn9bAfnbB79U7NdXRepUYInxi9TQnv/HF5haIY+2tWhE1W3WfdczvNexn7b1890vVjdErgAmNWvdLfsSmEUNhpor3R3P7mAVLJbcYFpHhXYcdVbyDneulE2GdPRrX9QptIS/BPSr7qh7FYk3fzCI6H/UXdP80++ezYL/1wsqPb8fQi7jPvOa/j5k9kGiAo/ailfrQlzp2XKHxjStW8GPPNbpe/BGOq+Q3PIqxoKuvcsdufrsYKDkfXLzJ2XfSSVaLtjMO7nnEjjK/sMlU+Sxvg9JSqm1jhR36rM/+H7+7vFb8pdfyy7P122urt9c+u7P88T39K7A/Vn+u/hJ0saY+VB+rh2CThyDTP6p/Vf+h/vP8H87/5fzfzv+ds/7wBxrzM5X77/y//w+axVEV</latexit> <latexit 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</latexit> 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</latexit> <latexit 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</latexit> [Liereo, Mielke, Savar´ e 2015] [Chizat, Schmitzer, Peyr´ e, Vialard 2015] Balanced OT Unbalanced OT min ⇡ R cd⇡+ <latexit 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<latexit 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<latexit 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<latexit 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⇢KL(⇡1 |↵) + ⇢KL(⇡2 | ) <latexit 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<latexit 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<latexit 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<latexit 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[Solomon et al, SIGGRAPH 2015] Generalizations ↵1 <latexit 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<latexit 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↵2 <latexit 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<latexit 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min P k kWp p (↵k, ) <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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  25. 3. Sinkhorn Divergences <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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x <latexit 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<latexit 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d(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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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  26. Sinkhorn Divergences 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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" > 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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<latexit 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min ↵ Wp ",p (↵, ) <latexit 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<latexit 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<latexit sha1_base64="04xOFucC5Ut2/g96/R1TregIzEs=">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</latexit> W"(↵, ↵) 6= 0 <latexit 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<latexit 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<latexit 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  27. Sinkhorn Divergences 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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" > 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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<latexit 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min ↵ Wp ",p (↵, ) <latexit 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<latexit 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<latexit sha1_base64="04xOFucC5Ut2/g96/R1TregIzEs=">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</latexit> W"(↵, ↵) 6= 0 <latexit 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<latexit 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<latexit sha1_base64="4NYv/0KG4ajqp7qRl0BSlmTG76Y=">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</latexit> Wp p," (↵, ) def. = Wp p," (↵, ) 1 2 Wp p," (↵, ↵) 1 2 Wp p," ( , ) <latexit 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</latexit> <latexit 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</latexit>
  28. Kernel norms (MMD): <latexit 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<latexit 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<latexit

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||⇠||2 dp def. = R X2 d(x, y)pd⇠(x)d⇠(y) <latexit 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<latexit 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Sinkhorn Divergences 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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" > 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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<latexit 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min ↵ Wp ",p (↵, ) <latexit 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<latexit 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<latexit sha1_base64="04xOFucC5Ut2/g96/R1TregIzEs=">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</latexit> W"(↵, ↵) 6= 0 <latexit 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<latexit 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<latexit sha1_base64="4NYv/0KG4ajqp7qRl0BSlmTG76Y=">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</latexit> Wp p," (↵, ) def. = Wp p," (↵, ) 1 2 Wp p," (↵, ↵) 1 2 Wp p," ( , ) <latexit 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</latexit> <latexit 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Wp p (↵, ) <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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" ! +1 <latexit 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</latexit> 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</latexit> 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Wp ",p (↵, ) <latexit 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||↵ ||2 dp <latexit 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</latexit> <latexit sha1_base64="MlLd/zUeVzbSgV3WYCjWVJMMZuc=">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</latexit> Theorem: [Genevay, P, Cuturi, 2017] <latexit sha1_base64="S6neu/pbqziddYWFY49jSDHFJFY=">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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit sha1_base64="S6neu/pbqziddYWFY49jSDHFJFY=">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</latexit> Arthur Gretton Proposition: || · || ||·||p is a norm for 0 < p < 2. <latexit 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<latexit 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<latexit 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  29. Sinkhorn Divergences concave <latexit 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</latexit> <latexit 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</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> concave <latexit 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</latexit> 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S" (↵, ) "!+1 ! 1 2 h ↵ , C ? (↵ ) i; (4) 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. min ↵ Wp ",p (↵, ) <latexit 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S" (↵, ) "!+1 ! 1 2 h ↵ , C ? (↵ ) i; (4) 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. min ↵ Wp ",p (↵, ) <latexit 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<latexit sha1_base64="nYWupsQXdFV69lfCLDQSS6Yalhc=">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</latexit> Wp p," (↵, ) def. = Wp p," (↵, ) 1 2 Wp p," (↵, ↵) 1 2 Wp p," ( , ) <latexit 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</latexit> <latexit 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</latexit> <latexit sha1_base64="MMvpdcWTyNwp0Jp9lABlK/06LIw=">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</latexit> Theorem: <latexit sha1_base64="WCp6tVB6SGFNLVO3ywKAXDF/o30=">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</latexit> <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> W",p > 0 and Wp ",p (·, ) is convex. <latexit 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</latexit> <latexit 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</latexit> 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</latexit> 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W",p(↵n, ) ! 0 () ↵n weak⇤ * <latexit 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  30. Sample Complexity Theorem: n d = 2 d = 3

    Optimal transport: su↵ers from curse of dimensionality. ! Adapt to support dimensionality [Weed, Bach 2017] 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 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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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ˆ ↵ <latexit 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<latexit 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<latexit 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log(n) log(n) E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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log(Wp(ˆ ↵, ˆ ↵0)) <latexit 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<latexit 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  31. Sample Complexity Theorem: n d = 2 d = 3

    Optimal transport: su↵ers from curse of dimensionality. ! Adapt to support dimensionality [Weed, Bach 2017] 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 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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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ˆ ↵ <latexit 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<latexit 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<latexit 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log(n) log(n) E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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log(Wp(ˆ ↵, ˆ ↵0)) <latexit 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log(n) d = 5 log(n) d = 2 log(W",p(ˆ ↵, ˆ ↵0)) <latexit 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E(|W",p(ˆ ↵, ˆ) W",p(↵, )|) = O(" d 2 n 1 2 ) <latexit 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<latexit 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Theorem: [Genevay, Bach, P, Cuturi] <latexit 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<latexit 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<latexit 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  32. 3. Sinkhorn Divergences <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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x <latexit 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<latexit 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d(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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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  33. 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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  34. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <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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min ✓ c KL( |↵✓) def. = X i log(⇢✓(xi)) <latexit 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<latexit 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  35. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <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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min ✓ c KL( |↵✓) def. = X i log(⇢✓(xi)) <latexit 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<latexit 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<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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c KL( |↵✓) = +1 <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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min ✓ Wp ",p (↵✓, ) <latexit 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  36. 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) . . .)
  37. 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) . . .)
  38. 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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sha1_base64="PiJWHJFbS74WSpUZvh1mwt8Pg+M=">AAA94XictVvbbhy5EaU3l107N2/yGCDoROvADhxDUgwkwELA2pJsa621Zc9I9u62LcylNWq7Z3o8PSNfZucD8hbkNR+TD8gX5A+SpyCvyUuqimSTPcPuIhWvGpLYHJ5TxWqyWEX2dMdZWkzX1/9+4YPvfPd73//wo4uXfvDDH/34J5c//ulRkc8mveSwl2f55Gm3UyRZOkoOp+k0S56OJ0ln2M2SJ92X2/j5k7NkUqT5qD19O06eDTuDUXqS9jpTqDq+nMSzYTqax9PTRRTvJlehcC2Kk1f95CSKc0Ai8TwedvM38yeLxfE8TsZFmuWj6+PF8/HVeUwqzCdJfxF3smPkua4ru9ksWcTdZHHt+PLa+o11+olWCxuqsCbUz0H+8S82RSz6Ihc9MRNDkYiRmEI5Ex1RwPW12BDrYgx1z8Qc6iZQSunzRCzEJcDOoFUCLTpQ+xL+DuDua1U7gnvkLAjdAykZ/E4AGYkrgMmh3QTKKC2iz2fEjLV13HPiRN3ewv+u4hpC7VScQi2H0y19cdiXqTgRf6A+pNCnMdVg73qKZUZWQc0jq1dTYBhDHZb78PkEyj1CajtHhCmo72jbDn3+D2qJtXjfU21n4p+k5RW4ItFSvc9Lho44I/6InuYMPpP6ZCB5AAyJ6iOWXpOth9T7EbSfQ/0DuBZU0jbpwjWn2kUjchsuF3KbRd6Fy4W8yyL34XIh91nkAVwu5IFCInZCNnfjW3C58C1W8iO4XMhHLPIxXC7kYxZ5BJcLecQiv4LLhfyKRd6By4W8wyLvw+VC3meRbbhcyDaLPITLhTxkkbtwuZC7Clk/Uydw5cSTMrPyFpSrMtBTZFBzi9XvNnlHF/a2x5zu1WD5Wb0D/93YHQ+bJjXYXY9xd1KD5UfeXfCRbizvi+7RauLC3mOxezAC3Ng9Fvu5eFGD/dxjpr2swfJzbR/aubG89/0C7tzYL1jsAyi5sfwa9RBq3NiHHivGuAZ7wGIfiVc1WB+vP6nB8n6/BX7FjeXXqTa0d2N9vOmsBsv70yOIYNxYfrV6ArVu7BMW+1S8qcE+ZbFfgnd3Y7/0WGHf1WD1GnuJVpABxSMJzNgmtk45K7E0BrYOIz8r15aMYuMu1HOYQYkZEGbIIu6WiLueiP0Sse+tV1H60YLiXV5Kq0S0PBHdcm3C0pRt3y/bYynzQOyUiJ0lRFNEis9a9+WMogtdwyGn5cqFJZ8+5aX/xlKixkOz59WIhxWEHNunNPKvU7aEGRRaqonttFzjJTKi+ybEa8redC+1DB43Lb2CjXrDoroOVJdFvXWg3rKomQM1Y1FnDtQZizIz38bFHiPA2B+fxZzu5AiQMXL9FUFUcAtWnXswRyMYPwcQBT6mmofwv0W5N3c1aYbZPK6TuMvxrOKJJ1CaizWoN1nhDuXXGc2wBDSTLR+qHB/vcG9jruac9MKLciWPyh0Tf56U9BmUPBgtRjSfwnjuU82CojtZCsPfK+e9LoXhd8niC4riZSkMP1XaT8+he1th2+fAtmA2jZX1TTmUQ+6/SA5dvkSrLnpcfKpDNWaQ700g/556MnvneC7bVJL2MeUwjsLqX1HpXwiHsXNh2TmMBaMnGfXqUhTck5HKe005VIecVtGR0sPchT4ZbNNXT0aXwzgOIOLappx7bpVDR++47I0ph3EcCbnvuaBIXpfDOAZ0L+1hymEcuNvSUXm+KYd6drSAzJ1NOdSrj2gXGPeA5JiXNSYqmlCcNFNsKcUHzbs1dsy/uo7hns3zMkdoZjKxbT1Pt1zLmjXS8UICXm0aqAfGFzMrBqtyzMUmm19JHaaV9X2Vx6zxaPl9sGIEs1+eAXB75hloqPck0HtnwLjBZl3VnmncJovDUXKyhIpV7ZSNFo1cuWtUrTumWi4vM701dozJXxc09sYUE+6TZTk77Nc+4TpGzkL7FQvxfCG2e6fma9X66yxuvIQYlyOtRydC8iStOU91Wb1l2fiKOuWZwiXPfMz4xd3mE+VtMOfJyRehLk0y7XZ6H8muw3X1ujB73PKziJ4o+qsz8hopnUgVbBaqd4tlND6ne8N9SGdyKENy9OA5RoplLOSpGe6i4356RB7V9recbLSX3qGT5YK8rvbHzeiBhR440OE5zjasGA+g1Iac4RDu2h5ZzqXSVjlZfCJ+W56O5vQEmzP6rOIhNYf0N0nFQzZl2acVlteAxtEgs3R/jmUejY9XmPis36WPyV2rnv8Kndzq8+0OjfH60Vy/E9MnqZskNaJZI0915d2yBKnB3PnJJsWvzb1EeSES0YdyUp9bkqVdRnTin1AGO6bIOKPZxs2Oamt7f2r5Ey3pQOizczzNzslDRuT/IlifchqTEf3a7w7oE3TpETLykT5+Jy2jG1esk7JjzMRxqZBvNZjxlpAvm5F8zWvProLGoswY5DqwWBrb2ib7FAsmJHWivLuZ282rDyLNexL2KJGMZqxcJfnX6K/+1eNkbWVEoIXxCRTK17meR045C9qoQ6t8sw/SbW0tPyl1eK60Nuuf0emTimY7lHGhPrha90Fyj+6lLBwlE9K7WGkj19Gm3VxkHi/ZEXt7Qlm89PsDtQKj3tdplVyjORfTKBnAKJiWWYRuy+0iL8ttllVl9+MuvhV2Y+uq1ZAxEmYHV1qI299PKFuztcxgVMvx+5Jmk9vqk6VWzXJGNBaH1lz+Bmp/CX+13vrej6db8Qq3aQxIBnNnLCJropUWfrJuV2Tpkam5zL2RZ8akbmXXnCe/lt7N5NhnwSwHNGreqF0LXT4PxwuL44WnDdt01misqOu1Jzpmc4u2Oq30lRcirR3APGOZ+YhMo1IPLe1cyo+1z7LyOb5GvWO51lmuDsxW+zTAnvM+SPdcX57d35SreyTuUGzTowhM5i99mqUpxVy6tjlTkwwo+abyr/bsj6kGpXfJgyKzfI8TZ4w8derRtSg1/bVa2XLy88Yj6PeWXqs22sfGVP7dCnJIc6KgeakRN6lFovS39YiWPNINK+aIaOe/QzGVjDuac2a7tXkmUSWeMPmmnFVGlswURmR/budtbyV73bPy14hywpmKrrvAFf6EkUFi9E6CO7Is6AnhKidPEmRE2yX/ueqn5CneyNLoBmk9F1sePkZmvWas22NL91j37TfQEq1unrqrBS8v85bIyTvPiV6HVrWhilHnS/fn4+qoVa5632SH2ZJcY48ZtbEzC5PlVTGx+NRbitQoTIrE+EgJ60WI/mGah+gsT6d8mXVrzVzdaZA+5pTyJe49UES4orurzmjuGtOP7gpfl7A2m6zhmHA3Llf7A7anxV2piyvrkKy92LgaZdZKVLdSaHZ7tTD+W3rIhLxfJrg9G9na1j2uZCn8Loxk6An5Rm9dfmhzfgoX/o2EKzvUEn32DlsQ394S22L3PbwN8UqV5Y5mRDXoC/pLuXdH9bPaotlGryx2m99Hgr+MFGzNaZ/SShqqu2TmNbfZ/flfkxeYiITV3rQM74Mthe/JqqSQ/qTk2fjepEJ/Fye0L1qCT0+qUvzlyHMNrhcnQn+nKawPmp3vQVVCiAz9HoPfMzetw2XZkprttSrFV4ZcBfSJi8bhyV99rmLa+XioifVE3r8E9A4nDex6tfh/+6HlGEnhsnylFfRdsxceT122S9SOLMbD4XPGSPMZzfUS/WXmZe9MtOSWJ+O+KOhJ5VZv3j8/xqNmDGhZcyH3QXntJN4eRUZfXxY8F3DpkIt/ib9e4L+N8KrkqNMjhEmfU9Sz6RY8m/7Gpat3+jMfnQxPnU5VNpNHtOiN2G2xJ+7A73YZAYa+HSq/Syn/I9b9/dk+1J6Q99C76HLnIKa6hHY/zClan+7VHuPx5bWN5W8hrxaONm9srN/YeHRz7bPb6hvKH4mfi19BXrIhfi8+E/egv4eg09/Ev8V/xH+3elt/3PrT1p9l0w8uKMzPROVn6y//A3hG/4c=</latexit> ˆ E(✓) def. = Wp ",p ( 1 m P i g✓(zi ), ) <latexit 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✓ ✓ ⌧r ˆ E(✓) <latexit 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  39. 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=">AAA9jHictVvbchu5EYU3t7VyWTt5TFVqEq1TuylHJSlOJSmXqta6a621ZZOSvbu0VbyMqLFHHJpDSra5/JS8Jp+R78gfJE/5hXQ3gAGGxEwDiqMpSRgQ53SjB2h0A8POME3y8erqP2989L3v/+CHP/r45tKPf/LTn31y6/bPT/JsMurGx90szUbPO+08TpNBfDxOxmn8fDiK2xedNH7Web2Fnz+7jEd5kg2a43fD+MVFuz9IzpJuewxVp7duT1tEMu2kk3jW6sSz01vLqyur9BMtFtZUYVmon6Ps9q/WRUv0RCa6YiIuRCwGYgzlVLRFDte3Yk2siiHUvRBTqBtBKaHPYzETS4CdQKsYWrSh9jX87cPdt6p2APfImRO6C1JS+B0BMhJ3AJNBuxGUUVpEn0+IGWuruKfEibq9g/8dxXUBtWNxDrUcTrf0xWFfxuJM/Jn6kECfhlSDvesqlglZBTWPrF6NgWEIdVjuwecjKHcJqe0cESanvqNt2/T5v6gl1uJ9V7WdiH+TlnfgikRD9T4rGNrikvgjepoT+Ezqk4LkPjDEqo9YuiJbX1DvB9B+CvWP4JpRSdukA9eUame1yC24XMgtFrkHlwu5xyIP4XIhD1nkEVwu5JFCInZENnfjG3C58A1W8hO4XMgnLPIpXC7kUxZ5ApcLecIiv4HLhfyGRe7C5ULussiHcLmQD1lkEy4Xsskij+FyIY9Z5A5cLuSOQlbP1BFcGfEkzKx8AOWyDPQUKdQ8YPXbJO/owm56zOluBZaf1dvw343d9rBpXIHd8Rh3ZxVYfuTtgY90Y3lftE+riQu7z2IPYAS4sQcs9kvxqgL7pcdMe12B5efaIbRzY3nv+xXcubFfsdhHUHJj+TXqMdS4sY89VoxhBfaIxT4RbyqwPl5/VIHl/X4D/Ioby69TTWjvxvp400kFlvenJxDBuLH8avUMat3YZyz2uXhbgX3OYr8G7+7Gfu2xwr6vwOo1dolWkD7FIzHM2Dq2djErsTQEtjYjPy3WlpRi4w7Uc5h+gekT5oJF7BWIPU/EYYE49NYrL/xoTvEuL6VRIBqeiE6xNmFpzLbvFe2xlHogtgvE9hyiLiLFZ637cknRha7hkONi5cKST5+ywn9jKVbjod7zasTjEkKO7XMa+XcpW8IMCi1Vx3ZerPESGdF9HeKKsjfdSy2Dx40Lr2Cj3rKojgPVYVHvHKh3LGriQE1Y1KUDdcmizMy3cS2PEWDsj89iSndyBMgYufqKICp4AKvOPszRCMbPEUSBT6nmMfxvUO7NXXWaYTaP6yTucrwoeeIRlKZiGepNVrhN+XVKMywGzWTLxyrHxzvc25iqOSe98KxYyaNix8SfJyF9+gUPRosRzacwnodUM6PoTpbC8PvFvNelMPwOWXxGUbwsheHHSvvxNXRvKmzzGtgGzKahsr4ph3LI/RfJoctLtOqix8WneqHGDPK9DeQ/UE/m4BrPZYtK0j6mHMaRW/3LS/0L4TB2zi07h7Fg9CSjXl2KgnsyUHmvKYfqkNEqOlB6mLvQJ4NteurJ6HIYxxFEXFuUc0+tcujoHRa9MeUwjhMh9z1nFMnrchhHn+6lPUw5jAN3W9oqzzflUM+OFpC5symHevUB7QLjHpAc87LGREUjipMmii2h+KB+t8aO+RfXMdyzeVnkCPVMJrat5ukUa1m9RjpeiMGrjQP1wPhiYsVgZY6pWGfzK6nDuLS+L/KYNR4tfwhWjGD2yzMAbs88BQ31ngR67xQY19isq9wzjVtncThKzuZQLVU7ZqNFI1fuGpXrTqmWy8tMb40dW+Svcxp7Q4oJD8mynB0OK59wFSNnocOShXi+ENu9V/O1bP1VFjecQwyLkdalEyF5klafp7qs3rBsfEed8ozhkmc+ZvzibvOZ8jaY82Tki1CXOpl2O72PZNfhunpXmD1u+VlETxT91SV5jYROpHI2C9W7xTIan9K94T6mMzmUITm68BwjxTIU8tQMd9FxPz0ij2r7W0422kvv0MlyTl5X++N6dN9C9x3o8BxnC1aMR1BqQs5wDHdNjyxnqbBVRhYfid8Xp6MZPcH6jD4teUjNIf1NXPKQdVn2eYnlCtA4GmSW7s8xz6PxrQUmPut36WNy17Lnv0Mnt/p8u01jvHo0V+/E9EjqOkmNaNbIU115Ny9BajB1frJO8Wt9L1FeiET0oZzUl5ZkaZcBnfjHlMEOKTJOabZxs6Pc2t6fmv9ESzoS+uwcT7Mz8pAR+b8I1qeMxmREv/a7A/oEXXqElHykj99JiujGFesk7BgzcVwi5FsNZrzF5MsmJF/z2rMrp7EoMwa5Dszmxra2ySHFgjFJHSnvbuZ2/eqDSPOehD1KJKMZK5+R/M/pr/7V42R5YUSghfEJ5MrXuZ5HRjkL2qhNq3y9D9JtbS0/LXR4qbQ265/R6dOSZtuUcaE+uFr3QHKX7qUsHCUj0jtfaCPX0brdXGQeztkRe3tGWbz0+321AqPed2mVXKY516JR0odRMC6yCN2W20Wel1svq8zux53/X9iNrctWQ8ZImB1caSFufz+mbM3WMoVRLcfva5pNbquP5lrVyxnQWLyw5vJ3UPtr+Kv11vd+PJ2SV9ikMSAZzJ2xiKyJFlr4ydosydIjU3OZeyPPjEndyq65Tn4tvZvJsS+DWY5o1LxVuxa6fB2OVxbHK08bNums0VhR12tPdMrmFk11WukrL0RaM4B5wjLzEZlGJR5a2rmUH2uPZeVzfI16z3KtslxtmK32aYA9532Q7rk+P7u/K1b3SOxSbNOlCEzmLz2apQnFXLq2PlOTDCj5nvKv9uxvUQ1K75AHRWb5HifOGHnq1KVrVmj6W7WyZeTnjUfQ7y1dqTbax7ao/IcF5AXNiZzmpUbcoxax0t/WI5rzSCtWzBHRzn+bYioZd9TnzHZr80yiUjxh8k05q4wsmSkMyP7cztvBQvZ6YOWvEeWEExVdd4Ar/Akjg8TonQR3ZJnTE8JVTp4kyIi2Q/5z0U/JU7yBpdEKaT0VGx4+Rma9ZqzbY0v3WPftd9ASrW6euqsFLy/1lsjJu86JXptWtQsVo07n7q/H1VarXPm+zg6TObnGHhNqY2cWJssrY1rivrcUqVGYFInxkRLWixD9wzQP0VmeTvky69aaubzTIH3MOeVL3HugiHBFd585o7nPmX50Fvg6hLXZZA3HhLtxmdofsD0t7krdXFiHZO3N2tUotVaiqpVCs9urhfHf0kPG5P1Swe3ZyNa27q1SlsLvwkiGrpBv9FblhzbnfbjwbyRc2aGW6LN32ID49oHYEjsf4G2IN6osdzQjqkFf0JvLvduqn+UW9TZ6Y7Hb/D4S/GUkYGtO+4RW0lDdJTOvuc3uz39FXmAkYlZ70zK8D7YUvieLkkL6k5Bn43uTCP1dnNC+aAk+PSlL8ZcjzzW4XpwJ/Z2msD5odr4HZQkhMvR7DH7P3LQOl2VLqrfXohRfGXIV0CcuGocnf9W5imnn46FG1hP58BLQO5zVsOvV4n/th5ZjJIXL8pWW03fNXnk8ddkuVjuyGA+HzxkjzWc0V0v0l5kVvTPRkluejPuioCeVWb358PwYj5oxoGVNhdwH5bWTeHsUGX19WfBcwKVDJv4j/nGD/zbCm4KjSo8QJn1OUc2mW/Bs+huXrt7pz3x0MjxVOpXZTB7RoDdit8SB2IXfrSICDH07VH6XUv5HrPv7sz2oPSPvoXfR5c5Bi+pi2v0wp2g9uld7jKe3ltfmv4W8WDheX/nLytqTe8tfbKovKH8sfil+A2nJmviT+ELsQ3ePQaUr8VfxN/H3jU82/rhxf2NDNv3ohsL8QpR+Nnb/C+XJ22s=</latexit> min ✓ E(✓) def. = Wp ",p (↵✓, ) <latexit 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<latexit 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sha1_base64="PiJWHJFbS74WSpUZvh1mwt8Pg+M=">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</latexit> ˆ E(✓) def. = Wp ",p ( 1 m P i g✓(zi ), ) <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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  40. 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?
  41. 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? Finite di↵erences: rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " 1) E(✓)) K(n + 1) operations, intractable for large n.
  42. 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? Seppo Linnainmaa This algorithm is reverse mode automatic di↵erentiation [Seppo Linnainmaa, 1970] Theorem: there is an algorithm to compute rE in O(K) operations. Finite di↵erences: rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " 1) E(✓)) K(n + 1) operations, intractable for large n.
  43. Examples of Images Generation g✓ X Z ↵✓ <latexit sha1_base64="lWnDUgue/pmzTJYmkYn0saVTkWY=">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</latexit>

    <latexit 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<latexit 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(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. digits in Figure 1c are more natural with smooth outline and sharper strike. For CIFAR-10 dataset, Inputs <latexit 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<latexit 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Generated ↵✓ <latexit 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  44. Examples of Images Generation g✓ X Z ↵✓ <latexit sha1_base64="lWnDUgue/pmzTJYmkYn0saVTkWY=">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</latexit>

    <latexit 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<latexit 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(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. digits in Figure 1c are more natural with smooth outline and sharper strike. For CIFAR-10 dataset, Inputs <latexit 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<latexit 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Generated ↵✓ <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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h⇠ <latexit 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<latexit 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<latexit sha1_base64="LDYKxMv5JRJeTkUU7AayQMWAocM=">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</latexit> Ian Goodfellow
  45. Progressive Growing of GANs for Improved Quality, Stability, and Variation

    Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018
  46. Progressive Growing of GANs for Improved Quality, Stability, and Variation

    Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018
  47. Conclusion: Toward High-dimensional OT Monge Kantorovich Dantzig Brenier Otto McCann

    Villani Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Style image (Y) S i J. Rabin Figalli