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A Schrödinger Tour of Data Sciences

Gabriel Peyré
September 16, 2021
1.4k

A Schrödinger Tour of Data Sciences

Talk at the Schrödinger's problem and Optimal Transport conference, Lisbon, 14-17 September 2021

Gabriel Peyré

September 16, 2021
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  1. A Schrödinger Tour of Data Sciences 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 work with: François-Xavier Vialard Lénaic Chizat Flavier Léger Pierre Roussillon
  2. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <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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  3. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Density fitting: Maximum likelihood (MLE) d↵✓(x) = ⇢✓(x)dx <latexit 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  4. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Density fitting: Maximum likelihood (MLE) d↵✓(x) = ⇢✓(x)dx <latexit 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g✓ X Z ⇣ Generative model fit: ! MLE undefined. ! Need a weaker metric. ↵✓ = g✓,]⇣ <latexit sha1_base64="W/xAH4ZEbKVL5sYCOuVPDm3rS+Q=">AAA9vXictVttcxu3EYbTl8Tum9N+7EznWsWdJKNqJMWZtpPRjC3JL4oVWzYp24lpa47kiaJ94tF3pGSb4f/pv+g/6Ne2/6D91L/Q3QVwwJG4W0B1dSMJB+J5drEHLHaBY3ecDovJ+vo/L33wgx/+6McffnT5yk9++rOf/+Lqx798XGTTvJcc9rI0y5924yJJh6PkcDKcpMnTcZ7Ep900edJ9tYOfPzlL8mKYjdqTt+Pk+Wk8GA2Ph714AlVHV7dnHSKZ5Ul/3onTo87kZB5tRYOjGZRWO8VJnI/nkWr1LB90n8/WvlxdX137ct55l0zieXR0dWV9bZ1+ouXChiqsCPVzkH38m03REX2RiZ6YilORiJGYQDkVsSjgeiY2xLoYQ91zMYO6HEpD+jwRc3EFsFNolUCLGGpfwd8B3D1TtSO4R86C0D2QksJvDshIXANMBu1yKKO0iD6fEjPW1nHPiBN1ewv/u4rrFGon4gRqOZxu6YvDvkzEsfgT9WEIfRpTDfaup1imZBXUPLJ6NQGGMdRhuQ+f51DuEVLbOSJMQX1H28b0+b+oJdbifU+1nYp/k5bX4IpES/U+KxlicUb8ET3NKXwm9UlB8gAYEtVHLJ2TrU+p9yNoP4P6+3DNqaRt0oVrRrXzRuQOXC7kDou8A5cLeYdF7sPlQu6zyAO4XMgDhURsTjZ341twufAtVvJDuFzIhyzyEVwu5CMW+RguF/Ixi/wOLhfyOxZ5Gy4X8jaLvAeXC3mPRbbhciHbLPIQLhfykEXegsuFvKWQ9TM1hysjniEzK29CuSoDPUUKNTdZ/bbJO7qw2x5zuleD5Wf1Lvx3Y3c9bJrUYG95jLvjGiw/8u6Aj3RjeV90l1YTF/Yui92DEeDG7rHYr8XLGuzXHjPtVQ2Wn2v70M6N5b3vN3Dnxn7DYu9DyY3l16gHUOPGPvBYMcY12AMW+1C8rsH6eP28Bsv7/Rb4FTeWX6fa0N6N9fGm0xos708fQwTjxvKr1ROodWOfsNin4k0N9imL/Ra8uxv7rccK+64Gq9fYK7SCDCgeSWDGNrHF5azE0hjYYkZ+Wq4tKcXGXajnMIMSMyDMKYu4UyLueCL2S8S+t15F6UcLind5Ka0S0fJEdMu1CUsTtn2/bI+l1AOxWyJ2FxBNESk+a92XM4oudA2HnJQrF5Z8+pSV/htLiRoPzZ5XIx5UEHJsn9DIX6VsCTMotFQT20m5xktkRPdNiHPK3nQvtQweNym9go16w6K6DlSXRb11oN6yqKkDNWVRZw7UGYsyM9/GdTxGgLE/PosZ3ckRIGPk+iuCqOAmrDp3YY5GMH4OIAp8RDUP4H+Lcm/uatIMs3lcJ3GX43nFE+dQmokVqDdZ4S7l1ynNsAQ0ky0fqBwf73BvY6bmnPTC83Ilj8odE3+eIekzKHkwWoxoPoXx3KOaOUV3shSGv1vOe10Kw98ii88pipelMPxEaT+5gO5thW1fANuC2TRW1jflUA65/yI5dPkKrbrocfGpnqoxg3xvAvn31JPZu8Bz2aGStI8ph3EUVv+KSv9COIydC8vOYSwYPcmoV5ei4J6MVN5ryqE6ZLSKjpQe5i70yWCbvnoyuhzGcQAR1w7l3DOrHDp6x2VvTDmM47GQ+55ziuR1OYxjQPfSHqYcxoG7LbHK80051LOjBWTubMqhXn1Eu8C4ByTHvKwxUVFOcdJUsQ0pPmjerbFj/uV1DPdsXpQ5QjOTiW3rebrlWtaskY4XEvBqk0A9ML6YWjFYlWMmNtn8SuowqazvyzxmjUfL74MVI5j98gyA2zNPQUO9J4HeOwXGDTbrqvZM4zZZHI6S4wVUR9VO2GjRyJW7RtW6I6rl8jLTW2PHDvnrgsbemGLCfbIsZ4f92idcx8hZaL9iIZ4vxHbv1HytWn+dxY0XEONypPXoREiepDXnqS6rtywbX1OnPBO45JmPGb+423ysvA3mPBn5ItSlSabdTu8j2XW4rq4Ks8ctP4voiaK/OiOvMaQTqYLNQvVusYzGZ3RvuA/pTA5lSI4ePMdIsYyFPDXDXXTcT4/Io9r+lpON9tI7dLJckNfV/rgZPbDQAwc6PMfZgRXjPpTakDMcwl3bI8u5UtoqI4vn4g/l6WhGT7A5o08rHlJzSH+TVDxkU5Z9UmE5BzSOBpml+3Ms8mh8Z4mJz/pd+pjcter5r9HJrT7fjmmM14/m+p2YPkndJKkRzRp5qivvFiVIDWbOTzYpfm3uJcoLkYg+lJP6wpIs7TKiE/+EMtgxRcYpzTZudlRb2/tTi59oSQdCn53jaXZGHjIi/xfB+pTRmIzo1353QJ+gS4+Qko/08TvDMrpxxTpDdoyZOG4o5FsNZrwl5MumJF/z2rOroLEoMwa5DswXxra2yT7FgglJzZV3N3O7efVBpHlPwh4lktGMlU9J/mf0V//qcbKyNCLQwvgECuXrXM8jo5wFbRTTKt/sg3RbW8tPSh1eKK3N+md0+qSi2S5lXKgPrtZ9kNyjeykLR0lOehdLbeQ62rSbi8zjBTtib48pi5d+f6BWYNR7lVbJFZpzHRolAxgFkzKL0G25XeRFuc2yqux+3MX/hd3Yumo1ZIyE2cGVFuL29xPK1mwtUxjVcvy+otnktnq+0KpZzojG4qk1l7+H2t/CX623vvfj6Va8wjaNAclg7oxFZE201MJP1nZFlh6ZmsvcG3lmTOpWds1F8mvp3UyOfRbMckCj5o3atdDli3C8tDheetqwTWeNxoq6XnuiIza3aKvTSl95IdLaAcxTlpmPyDRq6KGlnUv5sfZZVj7H16h3LNc6yxXDbLVPA+w574N0z/XF2f19ubpH4jbFNj2KwGT+0qdZOqSYS9c2Z2qSASVfV/7Vnv0dqkHpXfKgyCzf48QZI0+denTNS01/r1a2jPy88Qj6vaVz1Ub72A6Vv1hCntKcKGheasR1apEo/W09ogWPtGbFHBHt/McUU8m4ozlntlubZxJV4gmTb8pZZWTJTGFE9ud23vaWstc9K3+NKCecqui6C1zhTxgZJEbvJLgjy4KeEK5y8iRBRrRd8p/Lfkqe4o0sjdZI65nY8vAxMus1Y90eW7rHum+fQ0u0unnqrha8vNRbIifvIid6Ma1qpypGnS3cX4wrVqtc9b7JDtMFucYeU2pjZxYmy6tiOuIrbylSozApEuMjJawXIfqHaR6iszyd8mXWrTVzdadB+pgType490AR4YruPnVGc58x/egu8XUJa7PJGo4Jd+MytT9ge1rclbq8tA7J2suNq1FqrUR1K4Vmt1cL47+lh0zI+6WC27ORrW3dO5Ushd+FkQw9Id/orcsPbc6v4MK/kXBlh1qiz95hC+Lbm2JH3HoPb0O8VmW5oxlRDfqC/kLuHat+Vls02+i1xW7z+0jwlzEEW3PaD2klDdVdMvOa2+z+/OfkBXKRsNqbluF9sKXwPVmWFNKfIXk2vjdDob+LE9oXLcGnJ1Up/nLkuQbXi2Ohv9MU1gfNzvegKiFEhn6Pwe+Zm9bhsmxJzfZaluIrQ64C+sRF4/Dkrz5XMe18PFRuPZH3LwG9w3EDu14t/td+aDlGUrgsX2kFfdfspcdTl+0StSOL8XD4nDHSfEZzvUR/mVnZOxMtueXJuC8KelKZ1Zv3z4/xqBkDWtZMyH1QXjuJt0eR0deXBc8FXDpk4j/ir5f4byO8Ljnq9Ahh0ucU9Wy6Bc+mv3Hp6p3+zEcnw1OnU5XN5BEteiN2R+yJ2/C7U0aAoW+Hyu9Syv+IdX9/tg+1x+Q99C663DnoUF1Cux/mFK1P92qP8ejqysbit5CXC4eba39e23h4feXGtvqC8kfi1+J3kJZsiD+KG+IudPcQVPqL+Jv4u/jH1o2t4610aySbfnBJYX4lKj9b5/8FoXTumg==</latexit> <latexit 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<latexit 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<latexit 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↵✓ <latexit 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min ✓ Wp ",p (↵✓, ) <latexit 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  5. Shrödinger and kernels dimensionality Curse of meets Shrödinger Gromov X

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convergence Sinkhorn and
  6. Static Schrödinger Problem Schr¨ odinger’s problem: [1931] Erwin Schrödinger <latexit

    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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit 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KL(⇡|↵ ⌦ ) def. = R X2 log ⇣ d⇡(x,y) d↵(x)d (y) ⌘ d⇡(x, y) <latexit 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(mutual information) <latexit sha1_base64="OMzXbMItV2dR9mIKv1CwPSTdQF0=">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</latexit> Relative entropy: <latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. =
  7. Static Schrödinger Problem Schr¨ odinger’s problem: [1931] Erwin Schrödinger ↵

    <latexit 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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit sha1_base64="1zohFdB3aQQ05l3cPqYNxZB6ULI=">AABFV3ictVxfcxPJER9fkgvx5Q8kj3nZiyFlUj5ifCSXq6tUHcjG+NCBQbLhDgG1klZiYaUV+odB6LPlc1zlOVXJUz5C0t0zszMrzW7POsRbtmZH8+vu6Z3p6e6ZdXuUxJPp7u4PGx/96Mc/+finF362+cnPf/HLX1289OvTSTobd6KTTpqk48ftcBIl8TA6mcbTJHo8GkfhoJ1Ej9qvavj9o3k0nsTpsDl9O4qeDsL+MO7FnXAKVc8vzi637ta3W6O49r4VJrWglU7jQTQJWu2odnWzFb3uRr1gsxUPp88Xrcdnz/aWQStJ+61RuAhavXHYWbS6AcK3z3beXl3CHZLZPrsaQAlobENlABir0eblzecXt3av7dJPsF64rgpbQv0cp5c+/Ydoia5IRUfMxEBEYiimUE5EKCZwPRHXxa4YQd1TsYC6MZRi+j4SS7EJ2Bm0iqBFCLWv4G8f7p6o2iHcI80JoTvAJYHfMSADcQUwKbQbQxm5BfT9jChjbRHtBdFE2d7CZ1vRGkDtVLyAWg6nW/risC9T0RN/oT7E0KcR1WDvOorKjLSCkgdWr6ZAYQR1WO7C92Modwip9RwQZkJ9R92G9P0/qSXW4n1HtZ2Jf5GUV+AKREP1Ps0ohGJO9AN6mjP4TsqTAOc+UIhUH7H0hnQ9oN4Pof0C6u/BtaSS1kkbrgXVLkuRNbhcyBqLPITLhTxkkXW4XMg6izyGy4U8VkjEjknnbnwDLhe+wXJ+AJcL+YBFPoTLhXzIIk/hciFPWeT3cLmQ37PI23C5kLdZ5F24XMi7LLIJlwvZZJEncLmQJyzyAC4X8kAhi2fqGK6U6MTMrLwJ5TwPtBQJ1Nxk5btF1tGFveUxpzsFWH5W78OnG7vvodOoAHvgMe56BVh+5B2CjXRjeVt0h1YTF/YOiz2CEeDGHrHYb8TLAuw3HjPtVQGWn2t1aOfG8tb3W7hzY79lsfeg5Mbya9R9qHFj73usGKMC7DGLfSBeF2B9rP64AMvb/QbYFTeWX6ea0N6N9bGmswIsb09PwYNxY/nV6hHUurGPWOxjcVaAfcxivwPr7sZ+57HCvivA6jV2k1aQPvkjEczYMmphNiuxNAJqIcM/ydaWhHzjNtRzmH6G6RNmwCIOM8ShJ6KeIereck0yOzohf5fn0sgQDU9EO1ubsDRl23ez9lhKPBD7GWJ/BVHmkeKz1n2Zk3ehazjkNFu5sOTTpzSz31iK1Hgot7wacT+HkGP7BY38HYqWMIJCTZVRe5Gt8RIZ0H0Z4g1Fb7qXmgePm2ZWwUadsai2A9VmUW8dqLcsauZAzVjU3IGasygz821cy2MEGP3js1jQnRwB0kcuvgLwCm7CqnMH5mgA4+cYvMCHVHMfPhsUe3NXmWQYzeM6iVmOpzlLPIbSQmxBvYkK9ym+TmiGRSCZbHlfxfh4h7mNhZpz0govs5U8yDIm/nRikqef0UFvMaD5VI3OXapZkncnS9Xwd7J5r0vV8Aek8SV58bJUDT9V0k/PIXtTYZvnwDZgNo2U9k25Kg2Zf5E0dHmTVl20uPhUB2rMIL2zivSP1JM5OsdzqVFJ6seUq9GYWP2b5PpXhYbR88TSczUq6D1Jr1eXgso9Gaq415SrypDSKjpUcpi7qk8G23TVk9HlajSOweOqUcy9sMpVR+8o640pV6NxKmTec0mevC5Xo9Gne6kPU65GA7MtoYrzTbmqZUcNyNjZlKta9SFlgTEHJMe8rDFe0Zj8pJmiFpN/UJ6tsX3+9XUMczbPshihnJLxbYvptLO1rFwi7S9EYNWmFeVA/2Jm+WB5Gguxx8ZXUoZpbn1fp2PWeNR8HbQYwOyXewBczjwBCXVOAq13AhSvs1FXvmcat8ficJT0VlAtVTtlvUXDV2aN8nXPqZaLy0xvjR5bZK8nNPZG5BPWSbOcHuqFT7iIIqehek5DPL0qunun5mte+7ssbrSCGGUjrUM7QnInrTxOdWm9Yen4itrlmcIl93zM+MVsc09ZG4x5UrJFKEsZT7udziPZdbiu7giT45bfBfRE0V7NyWrEtCM1YaNQnS2W3viC7g3tE9qTQx6SRgeeY6CojITcNcMsOubTA7Kotr3leKO+dIZOlidkdbU9Lkf3LXTfga4e49RgxbgHpSbEDCdw1/SIcjYzXaWk8bH4LNsdTekJlkf0Sc5CahrS3kQ5C1kWZb/IUXkDaBwNMkr3p7FKR+Nba5T4qN8lj4ld85b/Cu3c6v3tkMZ48WguzsR0iesecQ1o1shdXXm3ykFKsHB+s0f+a3kvkV8VjmhDOa7PLM5SL0Pa8Y8ogh2RZ5zQbONmR761nZ9a/UZzOhZ67xx3s1OykAHZvwDWp5TGZEC/9tkBvYMuLUJCNtLH7sSZd+PydWJ2jBk/LhbyVIMZbxHZshnx13Tt2TWhsSgjBrkOLFfGttZJnXzBiLiOlXU3c7t89UGkOSdhjxJJ0YyVbeJ/lf7qXz1OttZGBGoYn8BE2TrX80gpZkEdhbTKl9sg3daW8nImwzMltVn/jEyXc5LtU8SF8uBq3QXOHbqXvHCUjEnuyVobuY6WZXOR8mhFj9jbHkXx0u731QqMcu/QKrlFc65Fo6QPo2CaRRG6LZdFXuVbzitP3Y/25P9C3eg6rzWkGAiTwZUa4vL7EUVrtpQJjGo5fl/RbHJrfbzSqpzPkMbiwJrL76H2U/ir5db3fnTaOatwi8aApGDujEZkTbDWwo/XrRwvPTI1LXNv+JkxqVvZNeeJr6V1MzH2vDKVYxo1ZyprocvnofHSovHSU4dN2ms0WtT12hI9Z2OLptqt9OVXhVuzAuUZS5n3yDQq9pDSjqX8qHZZqnyMr1HvWFq7LK0QZqu9G2DPeR+ke66vzu732eoeiNvk23TIA5PxS5dmaUw+l64tj9QkBeR8Q9lXe/a3qAa5t8mCImV5jhNnjNx16tC1zCT9vVrZUrLzxiLoc0tvVBttY1tU/nwNOaA5MaF5qRE3qEWk5LflCFYs0jXL5wgo8x+STyX9jvKY2W5tnkmQ8ydMvClnleElI4Uh6Z/LvB2tRa9HVvwaUEw4U951G2hVf8JIQWJ0JsHtWU7oCeEqJ3cSpEfbJvu5bqfkLt7QkugaSb0Qf/WwMTLqNWPdHlu6x7pvf4CWqHXz1F0teH6JN0eO33l29EJa1QbKR12s3J+PVqhWufx9mR5mK3yNPmbUxo4sTJSXx7TEV95cpETVuEiMD5dqvagifzXJq8gsd6d8KevWmnI+0yBtzAuKl7hzoIhweXfbTm/uKtOP9hq9NmFtarKGo4TZuFTlB2xLi1mpYCVCsuu5NSmx1qOi9cLwsFcNY8elpYzICiaCy93I1nYfWrlohc/GSAodIU/2FsWJNs2v4MK/gXBFiZqjTw6xAX7uTVETBx/gVMRrVZaZzYBq0CZ0V2LwUPUz36JcR68t6jZ9Hw7+PGLQNSd9TCtqVdklZV5ym7o//TdkDcYiYqU3Lav3webC92SdU5X+xGTh+N7EQr+TU7UvmoNPT/Jc/PnI/Q2uFz2h322q1gdNne9BnkMVHvo8g98zN62r87I5letrnYsvD7kO6J0XjcMdwOKYxbTzsVBj64l8eA5oHXol1PVq8b/2Q/MxnKrz8uU2oXfOXno8ddkuUplZ9IurzxnDzWc0F3P055lmvTNek5uf9P+CSk8qtXrz4emjX2rGgOa1EDIfyksn8fYoMvL6UsH9AZcMqfi3+NsG/1bC64xGkRxVKOn9imJqugVPTb956eqd/s5HJkOnSKY8NRNPNOhkbE0cidvwW8s8wKqnROU7lfITse73aLtQ2yProbPpMoPQorqIsiBmN61L9+YcbZHEeKZXnvFtQg3uidepFs/73qP2eOa3metb8Zskcq5/K1LRzUUmq7t8Zl61oQf5HTiZC9Lv+wZ0pl5ms+QJtIHHHqM8RyUjJf3284IQXYoLVyVdEEKPljLKbSflNp1Jigpot3N969AIH6mdftx3wPP5YZZdCsQfqS5UqwOu1JxUxw6pnlBmoE3634UI7U9iBz53VNkt6fGapBN6BnmJzqzvyk+CLZ3jwrzNeIXyYDpTN1ftUorqze5heSZ2v5CLPPFeju+X4PuWlA16Wq8o7h6L8tzhrITmTMlk7+cOhc57Sj1gNBtm46M8fp6X8Jp79P9uIfquJekhyNKmbHtA+3ljopco3RyQ9PJcZXne9k6JtPqtTUnTnKw040CfkSzfE0jUuCue/fIcJJeriQro2HNdnsjkTovETkr8/Bx5nIYIPXrL99WnpxyVGSvJzONN5LmHLHMPOj1Gmh5Loc9KouwD/fePL+knkIUvbqjCl9ez//5xunft+p+vff5gb+vrW+r/gFwQvxW/E9uw9n0hvobxfyxOgNPfNzY2Njc+2f9h/z8HHx9ckE0/2lCY34jcz8Gl/wIvk2VP</latexit> KL(⇡|↵ ⌦ ) def. = R X2 log ⇣ d⇡(x,y) d↵(x)d (y) ⌘ d⇡(x, y) <latexit 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(mutual information) <latexit sha1_base64="OMzXbMItV2dR9mIKv1CwPSTdQF0=">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</latexit> Relative entropy: <latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. =
  8. Static Schrödinger Problem Schr¨ odinger’s problem: [1931] Erwin Schrödinger ↵

    " ↵ ⇡" <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 ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit 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KL(⇡|↵ ⌦ ) def. = R X2 log ⇣ d⇡(x,y) d↵(x)d (y) ⌘ d⇡(x, y) <latexit 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(mutual information) <latexit 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Relative entropy: <latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. =
  9. Kernel norms and MMDs " > 0 : in general,

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

    there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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" <latexit 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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit 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W" p (↵, ) def. = <latexit 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⇡(") "!0 ! ↵ ⌦ <latexit sha1_base64="iAvjwgilPahhMA7Jt0rycZ4xuuI=">AABE/XictVxLcxy3EYacl6O85OToyziUUlKKVihaieNyucoSl6Jo0dJKu6Rki5JqH7OrFYc7q32J0pqVyo9J5ZJKJacc8zvyA1KVnPIX0g9ggNnFTGMYhVPkYrD4uhs9QKO7gWF7lAwm042Nf5x751vf/s53v/fu98//4Ic/+vFPLrz304NJOht34v1OmqTjR+3WJE4Gw3h/Opgm8aPROG4dt5P4YftoC79/OI/Hk0E6bE5fj+Inx63+cNAbdFpTqHp24f1eOo4uHl0+WX995bMPu/T5dHQxag270bMLaxtXN+gnWi1c04U1pX/q6Xsf/FMdqq5KVUfN1LGK1VBNoZyolprA9VhdUxtqBHVP1ALqxlAa0PexOlXnATuDVjG0aEHtEfztw91jXTuEe6Q5IXQHuCTwOwZkpC4BJoV2Yygjt4i+nxFlrC2ivSCaKNtr+GxrWsdQO1XPoVbCmZahOOzLVPXUb6kPA+jTiGqwdx1NZUZaQckjp1dToDCCOix34fsxlDuENHqOCDOhvqNuW/T9v6gl1uJ9R7edqX+TlJfgilRD9z7NKLTUnOhH9DRn8B3LkwDnPlCIdR+x9Ip0fUy9H0L7BdTfheuUSkYnbbgWVHtaityCy4fcEpE7cPmQOyJyDy4fck9E1uHyIesaidgx6dyPb8DlwzdEzvfh8iHvi8gHcPmQD0TkAVw+5IGI/BouH/JrEXkLLh/yloi8A5cPeUdENuHyIZsich8uH3JfRG7D5UNua2TxTB3DlRKdgTArb0A5zwMtRQI1N0T5bpJ19GFvBszpTgFWntU1+PRjawE6jQuw2wHjrleAlUfeDthIP1a2RbdpNfFhb4vYXRgBfuyuiP1CvSjAfhEw044KsPJc24N2fqxsfb+EOz/2SxF7F0p+rLxG3YMaP/ZewIoxKsDWRex99bIAG2L1xwVY2e43wK74sfI61YT2fmyINZ0VYGV7egAejB8rr1YPodaPfShiH6mTAuwjEfsVWHc/9quAFfZNAdassedpBemTPxLDjC2j1spmJZZGQK0l8E+ytSUh37gN9RKmn2H6hDkWETsZYicQsZch9oLlmmR2dEL+rsylkSEagYh2tjZhaSq272btsZQEIGoZoraEKPNI8VmbvszJuzA1EnKarVxYCulTmtlvLMV6PJRbXoO4l0Pw2H5OI3+doiWMoFBTZdSeZ2s8IyO6L0O8oujN9NLwkHHTzCq4qBMR1fag2iLqtQf1WkTNPKiZiJp7UHMRZWe+izsMGAFW//gsFnTHI4B95OIrAq/gBqw6t2GORjB+6uAFPqCae/DZoNhbusokw2ge10nMcjzJWeIxlBZqDeptVFij+DqhGRaDZNzyno7x8Q5zGws959gKn2YreZRlTMLpDEiefkYHvcWI5lM1Oneo5pS8Oy5Vw9/O5r0pVcNvk8ZPyYvnUjX8VEs/PYPsTY1tngHbgNk00tq35ao0OP/CNEz5PK26aHHxqR7rMYP0TirS39VPZvcMz2WLSqwfW65GY+L0b5LrXxUaVs8TR8/VqKD3xF6vKUWVezLUca8tV5UhpVV0qOWwd1WfDLbp6idjytVo1MHj2qKYe+GUq47eUdYbW65G40Bx3vOUPHlTrkajT/esD1uuRgOzLS0d59tyVcuOGuDY2ZarWvUhZYExB8RjnmusVzQmP2mmqQ3IPyjP1rg+/+o6hjmbp1mMUE7J+rbFdNrZWlYukfEXYrBq04pyoH8xc3ywPI2F2hTjK5ZhmlvfV+nYNR41vwdajGD28x6AlDNPQEKTk0DrnQDFa2LUle+ZwW2KOBwlvSXUoa6dit6i5ctZo3zdM6qV4jLbW6vHQ7LXExp7I/IJ90izkh72Cp9wEUVJQ3s5Dcn0qujujZ6vee1viLjREmKUjbQO7QjxTlp5nOrTesPR8SW9yzOFi/d87PjFbHNPWxuMeVKyRShLGU+3nckjuXW4rq4rm+Pm7yJ6omiv5mQ1BrQjNRGjUJMtZm98QfeW9j7tySEPptGB5xhpKiPFu2aYRcd8ekQW1bW3Em/Ul8nQcXlCVtfY43J030H3PejqMc4WrBh3odSEmGEf7poBUc75TFcpaXysPsx2R1N6guURfZKzkIYG25s4ZyHLouznOSqvAI2jgaP0cBrLdAz+cIWSHPX75LGxa97yX6KdW7O/3aIxXjyaizMxXeK6SVwjmjW8q8t3yxxYgoX3m03yX8t7ifyqcEQbKnF96nBmvQxpxz+mCHZEnnFCs02aHfnWbn5q+RvDqa7M3jnuZqdkISOyfxGsTymNyYh+3bMDZgedLUJCNjLE7gwy78bn6wzEMWb9uIHiUw12vMVky2bE39B1Z9eExiJHDLwOnC6NbaOTPfIFY+I61tbdzu3y1QeR9pyEO0qYoh0rl4n/Ffprfs04WVsZEahhfAITbet8zyOlmAV11KJVvtwGmbaulBczGZ5qqe36Z2W6mJOsRhEXyoOrdRc4d+ieeeEoGZPck5U2vI6WZXOR8mhJj9jbHkXxbPf7egVGuddplVyjOXdIo6QPo2CaRRGmrZRFXuZbzitPPYz25P9C3eo6rzWkGCmbwWUNSfn9mKI1V8oERjWP3yOaTX6tj5dalfMZ0lg8dubyN1D7Afw1cpv7MDrtnFW4SWOAKdg7qxGuiVZahPG6meNlRqahZe8tPzsmTSu35izxNVs3G2PPK1Op06g50VkLUz4LjRcOjReBOmzSXqPVoqk3luiZGFs09W5lKL8q3JoVKM9EyrJHZlCDACndWCqMalekKsf4BvVGpLUh0mrBbHV3A9w5H4L0z/Xl2f1NtrpH6hb5Nh3ywDh+6dIsHZDPZWrLIzWmgJyva/vqzv5DqkHubbKgSJnPceKM4V2nDl2nmaS/0CtbSnbeWgRzbumVbmNs7CGVP1pBHtOcmNC8NIjr1CLW8rtyREsW6arjc0SU+W+RT8V+R3nM7La2zyTK+RM23uRZZXlxpDAk/UuZt92V6HXXiV8jigln2rtuA63qTxgpMMZkEvye5YSeEK5yvJPAHm2b7OeqneJdvKEj0VWSeqE+C7AxHPXase6OLdNj07dfQkvUun3qvhYyvySYo8TvLDt6LVrVjrWPuli6Pxutll7l8vdlepgt8bX6mFEbN7KwUV4ec6g+DebCElXjwpgQLtV6UUX+apJXkZl3p0Ipm9aGcj7TwDbmOcVL0jlQRPi8u8teb+6K0I/2Cr02YV1qXCNRwmxcqvMDrqXFrFS0FCG59dKalDjrUdF6YXm4q4a142wpY7KCiZJyN9za7cNhLlqRszFMoaP4ZG9RnOjS/BQu/BspX5RoOIbkEBvg595QW2r7LZyKeKnLnNmMqAZtQncpBm/pfuZblOvopUPdpR/CIZzHAHQtST+gFbWq7ExZltylHk7/FVmDsYpF6W3L6n1wucg9WeVUpT8DsnBybwbKvJNTtS+GQ0hP8lzC+fD+htSLnjLvNlXrg6Eu9yDPoQoPc54h7Jnb1tV5uZzK9bXKJZQHrwNm58XgcAewOGax7UIs1Nh5Im+fA1qHXgl1s1r8r/0wfCyn6rxCuU3onbMXAU+d28U6M4t+cfU5Y7mFjOZijuE806x31mvy82P/L6r0pFKnN2+fPvqldgwYXgvF+VBZOsa7o8jKG0oF9wd8MqTqP+rv5+S3El5mNIrkqELJ7FcUUzMtZGrmzUtf78x3ITJZOkUy5anZeKJBJ2O31K66Bb9bmQdY9ZQov1PJn4j1v0fbhdoeWQ+TTecMwiHVxZQFsbtpXbq352iLJMYzvXzGtwk1uCe+R7V43vcutcczv81c34rfJOG5/qVKVTcXmSzv8tl51YYe5HfgOBdk3veN6Ew9Z7P4BNpxwB4jn6PiSMm8/bwgRJfiwmVJF4Qwo6WMcttLuU1nkuIC2u1c3zo0wkd6px/3HfB8fivLLkXqV1TX0qsDrtSSVHWPVI8pM9Am/W9AhPZrtQ6f67rsl7S+IumEnkFeohPnu/KTYKfecWHfZrxEeTCTqZvrdilF9Xb3sDwTWyvkwifey/H9EnzfkbJBT+uI4u6xKs8dzkpozrRM7n7uUJm8J+sBo9lWNj7K4+d5Ca95QP/vFKLvOJLugCxtyrZHtJ83JnqJ1s02Sc/nKsvztrdLpDVvbTJNe7LSjgNzRrJ8TyDR46549vM5SClXExfQcec6n8iUTosMvJTk+TkKOA3RCuit3NeQnkpUZqIks4A3kecBsswD6PQEaXoihb4oibYP9N8/PqGfiAsfX9eFT65l//3jYPPqtd9c/ej+5trnN/X/AXlXva9+ri7D2vex+hzGf13tA6ffqz+qv6i/1n5X+0PtT7U/c9N3zmnMz1Tup/a3/wKG/Eks</latexit> for k(x, y) = d(x, y)p and <latexit sha1_base64="Gp6WNxgbw+WZiDhK1JeM+2KyYQw=">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</latexit> h↵, ik def. = R k(x, y)d↵(x)d (y) <latexit 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W" p (↵, )p "!0 ! h↵, ik <latexit 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Prop.:
  11. Kernel norms and MMDs " > 0 : in general,

    there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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" <latexit 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min ⇡1=↵,⇡2= R X2 d(x, y)pd⇡(x, y) + " KL(⇡|↵ ⌦ ) <latexit 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W" p (↵, ) def. = <latexit 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⇡(") "!0 ! ↵ ⌦ <latexit sha1_base64="iAvjwgilPahhMA7Jt0rycZ4xuuI=">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</latexit> for k(x, y) = d(x, y)p and <latexit sha1_base64="Gp6WNxgbw+WZiDhK1JeM+2KyYQw=">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</latexit> h↵, ik def. = R k(x, y)d↵(x)d (y) <latexit 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W" p (↵, )p "!0 ! h↵, ik <latexit 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Prop.: Kernel norms (MMD): <latexit 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||↵ ||2 k def. = h↵ , ↵ ik Arthur Gretton <latexit 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||↵ ||2 k = 1 n2 X i,i0 k(xi, xi0 ) + 1 m2 X j,j0 k(yj, yj0 ) 2 nm X i,j k(xi, yj) <latexit 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k must be: <latexit sha1_base64="wDht3jJeopX8B4AeU9IOoUl+dQ8=">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</latexit> conditionally positive <latexit 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universal <latexit 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xi ↵ <latexit 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yj
  12. Sinkhorn Divergences fl [Ramdas, Garc´ ıa Trillos, <latexit sha1_base64="7m40Coq0F3c+G3SXCEnv0V4h1zw=">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</latexit> <latexit

    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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="6lzXbTDPpUQChkRXAFYhOyQ6AzI=">AABE93ictVzbchy3EYWcm6Pc5KTylJdxaKVkl6JQtBLHcaXKEpeiaFESpV1SskVJtZfhcsThzmpnl7qs+S2pvKRSyVO+IN+RD0hV8pRfSF+AAWYXM41hFE6Ri8HidDd6gEZ3A8PeOE3y6erqP869841vfuvb33n3u+e/9/0f/PBHF9778V6ezSb9eLefpdnkUa+bx2kyinenyTSNH40ncfe4l8YPe0fr+P3Dk3iSJ9moM309jp8cd4ej5CDpd6dQ9ezCT9vJ6Ogwm4yiVgLthvGoH+e/e3ZhZfXKKv1Ey4WrurCi9M9O9t77/1T7aqAy1VczdaxiNVJTKKeqq3K4HquralWNoe6JmkPdBEoJfR+rU3UesDNoFUOLLtQewd8h3D3WtSO4R5o5ofvAJYXfCSAjdREwGbSbQBm5RfT9jChjbRXtOdFE2V7DZ0/TOobaqTqEWglnWobisC9TdaB+S31IoE9jqsHe9TWVGWkFJY+cXk2BwhjqsDyA7ydQ7hPS6DkiTE59R9126ft/UUusxfu+bjtT/yYpL8IVqbbufVZQ6KoToh/R05zBdyxPCpyHQCHWfcTSS9L1MfV+BO3nUH8XrlMqGZ304JpT7Wktch0uH3JdRG7C5UNuishtuHzIbRG5A5cPuaORiJ2Qzv34Nlw+fFvkfB8uH/K+iHwAlw/5QETuweVD7onIr+DyIb8SkTfh8iFvisjbcPmQt0VkBy4fsiMid+HyIXdF5AZcPuSGRlbP1AlcGdFJhFl5HcplHmgpUqi5Lsp3g6yjD3sjYE73K7DyrG7Bpx/bCtBpXIHdCBh3BxVYeeRtgo30Y2VbdItWEx/2lojdghHgx26J2C/U8wrsFwEz7agCK8+1bWjnx8rW9w7c+bF3ROxdKPmx8hp1D2r82HsBK8a4ArsjYu+rFxXYEKs/qcDKdr8NdsWPldepDrT3Y0Os6awCK9vTPfBg/Fh5tXoItX7sQxH7SL2qwD4SsV+CdfdjvwxYYd9UYM0ae55WkCH5IzHM2Dpq3WJWYmkM1LoC/7RYW1LyjXtQL2GGBWZImGMRsVkgNgMR2wViO1iuvLCjOfm7Mpd2gWgHInrF2oSlqdh+ULTHUhqAaBWI1gKiziPFZ236ckLehamRkNNi5cJSSJ+ywn5jKdbjod7yGsS9EoLH9iGN/MsULWEEhZqqo3ZYrPGMjOi+DvGSojfTS8NDxk0Lq+CiXomongfVE1GvPajXImrmQc1E1IkHdSKi7Mx3cfsBI8DqH5/FnO54BLCPXH1F4BVch1XnFszRCMbPDniBD6jmHny2KfaWrjrJMJrHdRKzHE9KlngCpblagXobFbYovk5phsUgGbe8p2N8vMPcxlzPObbCp8VKHhUZk3A6CckzLOigtxjRfGpG5zbVnJJ3x6Vm+FvFvDelZvgN0vgpefFcaoafaumnZ5C9o7GdM2DbMJvGWvu23JQG51+Yhimfp1UXLS4+1WM9ZpDeq4b0t/ST2TrDc1mnEuvHlpvRyJ3+5aX+NaFh9Zw7em5GBb0n9npNKWrck5GOe225qQwZraIjLYe9a/pksM1APxlTbkZjBzyudYq550656egdF72x5WY09hTnPU/JkzflZjSGdM/6sOVmNDDb0tVxvi03teyoAY6dbbmpVR9RFhhzQDzmucZ6RRPyk2aaWkL+QX22xvX5l9cxzNk8LWKEekrWt62m0yvWsnqJjL8Qg1WbNpQD/YuZ44OVaczVmhhfsQzT0vq+TMeu8aj5bdBiBLOf9wCknHkKEpqcBFrvFCheFaOucs8Mbk3E4Sg5WEDt69qp6C1avpw1Ktc9o1opLrO9tXrcJ3ud09gbk0+4TZqV9LBd+YSrKEoa2i5pSKbXRHdv9Hwta39VxI0XEONipPVpR4h30urjVJ/W246OL+pdnilcvOdjxy9mmw+0tcGYJyNbhLLU8XTbmTySW4fr6mVlc9z8XURPFO3VCVmNhHakcjEKNdli9sbndG9p79KeHPJgGn14jpGmMla8a4ZZdMynR2RRXXsr8UZ9mQwdl3OyusYe16OHDnroQTePcdZhxbgLpQ7EDLtw1wmIcs4XuspI4xP1y2J3NKMnWB/RpyULaWiwvYlLFrIuyj4sUXkJaBwNHKWH01ikY/D7S5TkqN8nj41dy5b/Iu3cmv3tLo3x6tFcnYkZENc14hrRrOFdXb5b5MASzL3frJH/Wt9L5NeEI9pQietThzPrZUQ7/jFFsGPyjFOabdLsKLd281OL3xhOO8rsneNudkYWMiL7F8H6lNGYjOjXPTtgdtDZIqRkI0PsTlJ4Nz5fJxHHmPXjEsWnGux4i8mWzYi/oevOrpzGIkcMvA6cLoxto5Nt8gVj4jrR1t3O7frVB5H2nIQ7SpiiHSuXiP+H9Nf8mnGysjQiUMP4BHJt63zPI6OYBXXUpVW+3gaZtq6UHxQyPNVS2/XPyvRBSbIWRVwoD67WA+Dcp3vmhaNkQnLnS214Ha3L5iLl8YIesbcHFMWz3R/qFRjlvkyr5ArNuX0aJUMYBdMiijBtpSzyIt96XmXqYbTz/wt1q+uy1pBipGwGlzUk5fdjitZcKVMY1Tx+j2g2+bU+WWhVz2dEY/HYmctfQ+378NfIbe7D6PRKVuEGjQGmYO+sRrgmWmoRxutGiZcZmYaWvbf87Jg0rdyas8TXbN1sjH3SmMoOjZpXOmthymeh8dyh8TxQhx3aa7RaNPXGEj0TY4uO3q0M5deEW6cB5ZlIWfbIDCoJkNKNpcKoDkSqcoxvUG9EWqsirS7MVnc3wJ3zIUj/XF+c3V8Xq3ukbpJv0ycPjOOXAc3ShHwuU1sfqTEF5HxN21d39u9TDXLvkQVFynyOE2cM7zr16TotJP2FXtkysvPWIphzSy91G2Nj96n88RLymOZETvPSIK5Ri1jL78oRLVikK47PEVHmv0s+Ffsd9TGz29o+k6jkT9h4k2eV5cWRwoj0L2Xetpai1y0nfo0oJpxp77oHtJo/YaTAGJNJ8HuWOT0hXOV4J4E92h7Zz2U7xbt4I0eiKyT1XP0+wMZw1GvHuju2TI9N3z6Clqh1+9R9LWR+aTBHid9ZdvS6tKodax91vnB/NlpdvcqV7+v0MFvga/UxozZuZGGjvDJmX30WzIUlasaFMSFcmvWiifzNJG8iM+9OhVI2rQ3lcqaBbcwhxUvSOVBE+Ly7S15v7kOhH70lej3CutS4RqKE2bhM5wdcS4tZqWghQnLrpTUpddajqvXC8nBXDWvH2VLGZAVTJeVuuLXbh/1StCJnY5hCX/HJ3qo40aX5GVz4N1K+KNFwDMkhtsHPva7W1cZbOBXxQpc5sxlRDdqEwUIM3tX9LLeo19ELh7pLP4RDOI8EdC1Jn9CK2lR2pixL7lIPp/+SrMFExaL0tmXzPrhc5J4sc2rSn4QsnNybRJl3cpr2xXAI6UmZSzgf3t+QenGgzLtNzfpgqMs9KHNowsOcZwh75rZ1c14up3p9LXMJ5cHrgNl5MTjcAayOWWy7EAs1cZ7I2+eA1uGghrpZLf7Xfhg+llNzXqHccnrn7HnAU+d2sc7Mol/cfM5YbiGjuZpjOM+s6J31mvz82P+LGj2pzOnN26ePfqkdA4bXXHE+VJaO8e4osvKGUsH9AZ8MmfqP+vs5+a2EFwWNKjmaUDL7FdXUTAuZmnnz0tc7812ITJZOlUxlajaeaNPJ2HW1pW7C73rhATY9JcrvVPInYv3v0Q6g9oCsh8mmcwZhn+piyoLY3bQB3dtztFUS45lePuPbgRrcE9+mWjzve5fa45nfTqlv1W+S8Fy/ozI1KEUmi7t8dl71oAflHTjOBZn3fSM6U8/ZLD6Bdhywx8jnqDhSMm8/zwkxoLhwUdI5IcxoqaPc81Lu0ZmkuIJ2r9S3Po3wsd7px30HPJ/fLbJLkfoV1XX16oArtSTVjkeqx5QZ6JH+VyFC+7W6DJ+Xddkv6c6SpDk9g7JEr5zv6k+CnXrHhX2b8SLlwUym7kS3yyiqt7uH9ZnYViUXPvFejx/W4IeOlG16WkcUd09Ufe5wVkNzpmVy93NHyuQ9WQ8YzXaL8VEfP5/U8DoJ6P/tSvRtR9JNkKVH2faI9vMmRC/Vutkg6flcZX3e9laNtOatTaZpT1bacWDOSNbvCaR63FXPfj4HKeVq4go67lznE5nSaZHES0men+OA0xDdgN7KfQ3pqURlJkoyC3gT+SRAlpMAOgeCNAcihaEoibYP9N8/PqWfiAufXNOFT68W//1jb+3K1d9c+fj+2srnN/T/AXlX/Uz9XF2Cte8T9TmM/x21q/BUyB/VX9RfW69bf2j9qfVnbvrOOY35iSr9tP72X0TOSQA=</latexit> Sinkhorn Divergences:
  13. Sinkhorn Divergences fl [Ramdas, Garc´ ıa Trillos, <latexit sha1_base64="7m40Coq0F3c+G3SXCEnv0V4h1zw=">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</latexit> <latexit

    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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Sinkhorn Divergences: " ! +1 <latexit 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It6RhR9+nVRCj3xenLrypXmSglHdY/UlnqsDqCNb6q3jep4XnqmZJd+quwbr7iWTMpCvU/4kP+akI5iaa5aNH1UhxDfF+86OwR5sNaXrFqy5w7Kuc/IVtgHSv56WeuodhXPzcgphHIdchzf/EGpeA6J5915HIy7NfmkSvFmuKwgn8yElGWzS1rU+h1oO6x3V7vuEifpN+wVvb6aWdZeLUERactobsZY0MpLGW1omJzluqq3Vpc2yjfjNbUw225lpaqprX1fLO3/u53Vyb/skP/btDCOrJr5sls1FvYr2/nftp238VC33sqCruurfmVL/1dt6bp+yLYj/y6MPtUw3+jlOm+SlW6Z6+hb5uTurdA4sMhB9rynVszUslJlxd+skBXxxVjEt8f1evxMdNX0TuaYMTifOJC7Ns2duEO9AsXxW5fmGcSeFxTPzah18B28HSWnYD9QcjdyPXJSQd0vRMdlTIe4zUijZeT7+XiyeB8p7ovGGQT79IbMHib6JBbeToeWmhSQd3N6Yl8humVqfHu3zN5NShrm0x3mBuGkUL5iXtlry9FjkktkZJSd0xe5dImn3LwvSqQS6kX+fDdym3Ii9Ts5fVs2I0GPVizrdJWQVfu0aU683FO893xAdcwnUOYKd9XOHRzrOb09hzbphOexbD8q8wZlDZv8Vc/LFNmPjskzxFA0+asU62aRyuXG9nFPe+ekFuXWpo2t51hXp1dW/BziHENDtOimE6otpLQWRSlUS0VKLm+c6HpEWfrUyrtklbxvb0Zjhkz7Wd4xhNoN36N+VZvvsOAT+a7/EekRd0N0dJnreJRz1p/z4PMVM0KZd3MGT9rYQ3VMO3cTPRuFc70XnlIOabZtpvp6HUc0u6seqcfqz4FWH3J/7S3FTOcRbFLZj+XfT7Gf3+Zn7xAO3460rp7ndzTW7SOUdx6lPNf3vPnpbgWoVulu5XT9lEPy+iT2U36kntG4oYk2WOrjiB3Fj+nuNR9lvGnumHK1Kjfk19N9CDI8qtS9pIZ2JMleKb4JWqSxbwO8QScI+d4FtMNZI+k6lBbHxeS8Dqc9PXPrOr9lvgvpckfrsqdvAfLlvtC7IkN7IS+oFJzzlZJ9udMg/UzJeUwZY1/Q2WwZUZt4IUyleGbInKTkMtg0ZdweonuRy+ezac7Dd54tVPk+MZelx/CUO5FYH0WqZQ3HUNsu0GO75b3ZZYnjqTPtvbzkhnaR4krObUHliqvPPTWx6K4A8makPYl/K9tADFc+V2R/XrF0dUm3ddXpLiyf/GbQmG5PNNZaZ1t4+6KrNTTxGobnVUWGRW5nZauNo/jASdPW15XX1sK7Setllv7IR+EA+hq+8fggou/inKafwrtEi60hPNYQKut05uLQQeMoSOVBTgUtoK8pFekcRZSex6bFc0fuM9n+06qd/K5VvP30mGrgduPa3FPyWw3mE37vo+enyDu3eUwrbb5nnfJd0xESxzfmlBSe/7hDaS/hYSs91mfBE/XTiiQ2ek4jj1fqlkpyn+KvA74TSnpSO23FooTjq5tR9mVb19NIazK25M+7kefdUXE3HCJCMPZtA0UK8dLuwqjOTY+lj6d0oOmEWv5uzq84opoQt6cB9Oeq7q7Zz9WVszcUi+H5IvkFIPvXeSaEMKdZmHcxP/5iEKYMwZJHFP/OVbNfb0IKvJMHUZv01te7ftg3jGn0jBGa3Alhfn+IT59jS8NZK//ZNDx1PlauWeLyb5HwvA+fzUEp+ZzZIZRpTGfdP9Qe6yjCO4yIQvl2HJtDh+aYeGcBjp0fqb8JUn1NummXTmaF5H+ao5JSGUJnINBb81mDED+7NIfQLm1slavc/rQG9rMNfq/Zqame1qvEEOETq6cF+Y0vNrdAHGtv1Ymo2br7rGN+r2Evb+/Vu1/qboxeAUxo1LpX8SM2jRgKc1W+P5rbxzxYKrnFsIwM7zrsqfId7FwvvQjrHNC4blBqC0UJHlDZV/UoFmv6dh7R+ai/pPun2T+fBfutl1Z+fDuGXsR95rX4fczsgUQDHLWXrdSHvtSx4wqNb1yxgh97rtHN4o9wXCW/4VGOBV19lTt289vFSMn54PJNzr6TTrJatJlzcM8j9pT5hU2mymd5W5SWUm0bK+zRZ332//jd5bXyL71WX17cu7u2enft0/vLP/9r/SuwP1J/pP4UdLGmPlA/V0/AJg9Apr9X/6j+Vf3b+d+d/8P5P5//C2f94Q805g9V4b/zf/8frhtERQ==</latexit> <latexit 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</latexit> 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</latexit> 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</latexit> <latexit 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</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[L´ eonard 2012] <latexit 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<latexit 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<latexit sha1_base64="FktZ8qWo0YQDE1fIVBPQAYcO+ow=">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</latexit> Theorem: <latexit 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<latexit 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<latexit 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</latexit> 1 2 ||↵ ||2 dp
  14. Sinkhorn Divergences fl [Ramdas, Garc´ ıa Trillos, <latexit sha1_base64="7m40Coq0F3c+G3SXCEnv0V4h1zw=">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</latexit> <latexit

    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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Sinkhorn Divergences: " ! +1 <latexit 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</latexit> <latexit 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</latexit> 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</latexit> 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</latexit> <latexit 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</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[L´ eonard 2012] <latexit 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<latexit 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<latexit sha1_base64="FktZ8qWo0YQDE1fIVBPQAYcO+ow=">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</latexit> Theorem: <latexit 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<latexit 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</latexit> 1 2 ||↵ ||2 dp <latexit sha1_base64="uo2R6Eh4KXo54BjWqg9RXWBYUVA=">AABB/nictVxLcxvHER45L0t5WE6OuWxCKSWlZIWiVXEcx1WmSIqiCUuQAFKyDUm1AJbgigsshAX4EMyq/JYcckvlmluuuSa3/IPklL+QfszszAKzO7OMwikSs7Pzdff0zvR09yzYHSdxNl1d/eeld7717e9893vvXr7y/R/88EfvXX3/x/tZOpv0or1emqSTZ90wi5J4FO1N42kSPRtPonDYTaKn3aMNvP/0OJpkcTpqT8/G0fNhOBjFB3EvnELTy6u/60yjU8DNd6OzYDxJATf87XlwchiNgjgLrh3dOL11dvPTD/r0+WJ8LQiD2ShGkmHy8urK6u1V+gmWK3dkZUXIn2b6frAnOqIvUtETMzEUkRiJKdQTEYoMytfijlgVY2h7LubQNoFaTPcjcS6uAHYGvSLoEULrEfwdwNXXsnUE10gzI3QPuCTwOwFkIK4DJoV+E6gjt4Duz4gytpbRnhNNlO0MPruS1hBap+IQWl041dMXh2OZigPxGxpDDGMaUwuOriepzEgrKHlgjGoKFMbQhvU+3J9AvUdIpeeAMBmNHXUb0v1/UU9sxeue7DsT/yYpr0MJREuOPs0phOKY6Af0NGdwj+VJgPMAKERyjFg7IV0PafQj6D+H9odQzqmmdNKFMqfW80rkBhQbcsOJ3IZiQ247kQ0oNmTDiWxCsSGbEonYCencjm9BseFbTs6PodiQj53IJ1BsyCdO5D4UG3LfifwKig35lRN5H4oNed+J3IViQ+46kW0oNmTbidyDYkPuOZFbUGzILYksX6kTKCnRiR2rch3qRR5oKRJoWXfKd4+sow17z2NN90qw7lW9CZ927KaHTqMS7JbHvDsowbpn3jbYSDvWbYse0G5iwz5wYndgBtixO07s5+JVCfZzj5V2VIJ1r7UG9LNj3db3C7iyY79wYh9CzY5171GPoMWOfeSxY4xLsE0n9rF4XYL1sfqTEqzb7rfArtix7n2qDf3tWB9rOivBuu3pPngwdqx7t3oKrXbsUyf2mTgtwT5zYr8E627Hfumxw74pwao99grtIAPyRyJYsVXUwnxVYm0M1EIH/yTfWxLyjbvQ7sIMcsyAMEMnYjtHbHsiGjmi4S1XltvRjPxdN5dWjmh5Irr53oS1qbN/P++PtcQDsZkjNhcQVR4pPms1lmPyLlSLCznNdy6s+Ywpze031iI5H6otr0I8KiB4bh/SzL9F0RJGUKipKmqH+R7PyICuqxAnFL2pUSoebtw0twom6tSJ6lpQXSfqzII6c6JmFtTMiTq2oI6dKL3yTVzHYwZo/eOzmNMVzwD2kctLAF7BOuw6D2CNBjB/muAFPqGWR/DZotjbVaokw2ge90nMcjwvWOIJ1OZiBdp1VLhJ8XVCKywCybjnIxnj4xXmNuZyzbEVPs938iDPmPjTiUmeQU4HvcWA1lM9OrvUck7eHdfq4R/k617V6uG3SOPn5MVzrR5+KqWfXkD2tsS2L4BtwWoaS+3rel0anH9hGqp+hXZdtLj4VIdyziC905r0d+ST2bnAc9mgGutH1+vRyIzxZYXx1aGh9ZwZeq5HBb0n9npVLag9kpGMe3W9rgwp7aIjKYe+qvtksE9fPhlVr0ejCR7XBsXcc6Ned/aO89Hoej0a+4Lznufkyat6PRoDumZ96Ho9GphtCWWcr+t1LTtqgGNnXa9r1UeUBcYcEM95btFe0YT8pJmkFpN/UJ2tMX3+5X0MczYv8hihmpL2bcvpdPO9rFoi5S9EYNWmNeVA/2Jm+GBFGnOx5oyvWIZpYX9fpqP3eNR8A7QYwOrnMwBXzjwBCVVOAq13AhTvOKOu4sgUbs2Jw1lysIDqyNap01vUfDlrVGx7Sa2uuEyPVuuxQ/Y6o7k3Jp+wQZp16aFR+oTLKLo01ChoyE2vju7eyPVa1P6qEzdeQIzzmdajEyE+SauOU21abxk6vi5PeaZQ+MxHz1/MNh9Ia4MxT0q2CGWp4mn2U3kksw331VtC57j5XkBPFO3VMVmNmE6kMmcUqrLF7I3P6VrT3qMzOeTBNHrwHANJZSz41Ayz6JhPD8iimvbWxRv1pTJ0XM/I6ip7XI0eGOiBBV0/xtmAHeMh1NoQM+zBVdsjyrmS6yoljU/EB/npaEpPsDqiTwoWUtFgexMVLGRVlH1YoHICaJwNHKX701iko/CdJUruqN8mj45di5b/Op3cqvPtkOZ4+Wwuz8T0iesacQ1o1fCpLl8tcmAJ5tY7a+S/Vo8S+dXhiDbUxfWFwZn1MqIT/4gi2DF5xgmtNtfqKPY281OLdxSnplBn53ianZKFDMj+BbA/pTQnA/o13x1QJ+hsERKykT52J869G5uvEzvnmPbjYsFvNej5FpEtmxF/RddcXRnNRY4YeB84X5jbSicN8gUj4jqR1l2v7erdB5H6PQlzljBFPVduEP+b9Ff9qnmysjQjUMP4BDJp62zPI6WYBXUU0i5fbYNUX1PKa7kML6TUev/TMl0rSLZJERfKg7t1Hzj36Jp54SyZkNzZUh/eR6uyuUh5vKBHHO0BRfFs9wdyB0a5b9EuuUJrrkOzZACzYJpHEaqvK4u8yLeaV5G6H+3s/0Jd67qoNaQYCJ3BZQ258vsRRWumlAnMap6/R7Sa7FqfLPSq5jOiuTg01vI30Poz+KvkVtd+dLoFq3CP5gBT0FdaI9wSLPXw43WvwEvNTEVLX2t+ek6qXmbLReJrtm46xj6uTaVJs+ZUZi1U/SI0Xhk0XnnqsE1njVqLql1ZopfO2KItTyt9+dXh1q5Beeak7PbIFCr2kNKMpfyo9p1U3TG+Qr1x0lp10gphtZqnAeaa90Ha1/ri6v4m390DcZ98mx55YBy/9GmVxuRzqdbqSI0pIOe70r6aq79DLci9SxYUKfN7nLhi+NSpR+U8l/QXcmdLyc5ri6DeWzqRfZSN7VD9wyXkkNZERutSIe5Sj0jKb8oRLFik24bPEVDmPySfiv2O6pjZ7K2fSVDwJ3S8yatK8+JIYUT6d2Xedpai1x0jfg0oJpxJ77oLtOo/YaTAGJVJsHuWGT0h3OX4JIE92i7Zz2U7xad4I0Oi2yT1XHzqYWM46tVz3ZxbasRqbL+Enqh1/dRtPdz8Em+OLn4XOdELaVcbSh91vnB9MVqh3OWK11V6mC3w1fqYUR8zstBRXhHTEZ94c2GJ6nFhjA+XeqOoI389yevIzKdTvpRVb0W5mGlgG3NI8ZLrPVBE2Ly7G1Zv7qZjHN0lel3CmtS4xUUJs3GpzA+YlhazUpeX9iFuvVy5GyXGTlS2Uyjq5m6h7TdbyIisXyJcORvubcreKUQp7iwMU+gJfqO3LD40aX4CBf8GwhYdKo4+ucMW+LfrYkNsvYW3IV7LOmc0A2pBW9BfiL1DOc5ij2odvTaom/R9OPjziEHXLulj2knrys6U3ZKb1P3pn5AVmIjIKb3uWX8MJhf3SJY51RlPTJbNPZpYqO/i1B2L4uAzkiIXfz58ruEaxYFQ32mqNwZF3T2CIoc6PNR7DH7PXPeuz8vkVK2vZS6+PHgXUCcuCocnf+Wxiu7nY6EmxhN5+xzQOhxUUFe7xf86DsVHc6rPy5dbRt81e+Xx1LlfJDOy6A/XXzOam89sLufozzPNR6e9JTs/9vuCWk8qNUbz9umjP6rngOI1F5wHdUvHeHMWaXl9qeC5gE2GVPxH/PWS+9sIr3MaZXLUoaTOKcqpqR5uauobl7bRqXs+Mmk6ZTIVqek4okVvxG6IHXEffjdyD7Du26H8XUr+RKz9+7N9aD0g66Gy6Jw56FBbRNkPfYrWp2v9/myZxPguL7/b24YWPAtvUCu+5/uQ+uO7vu3C2Mq/QcJr/QuRin4hIlk83dPrqgsjKJ68cQ5Ifc83oHfpOYvFb54NPc4W1ftTixLN6Y77zYJuKb5rSNmjuTqWZ/V4coBv2Id5figQv6K2UNp53HNdnJulnJsLnDPSTpHDqXGv+t2sMi4bBpd+njs7lv1SirP1eV51bnSzlAu/g16NH1TgB4aULdL+EUXCE1GdzZtV0JxJmcwT1pFQmUjWA8aZYf68qyPb4wpexx7j3y1F7xqSboMsXcp/B3TCNiF6idTNFknPbzpWZ1IfVEgrv0dJ/93gY/oJuPLRXVn5+E7+3w32127f+fXtDx+vrXx2T/6fg3fFT8XPxQ1Y4x+Jz4BaU+wBhz+Iv4m/i3+s/379j+t/Wv8zd33nksT8RBR+1v/yX5pZuDY=</latexit> Key problem: when is k(x, y) = d(x, y)p a universal <latexit sha1_base64="HxAKm25RPr6nAbmkFGHVik0hUcY=">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</latexit> conditionaly positive kernel? Proposition: || · || ||·||p is a norm for 0 < p < 2. <latexit 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<latexit 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<latexit sha1_base64="RAaLm1CEm2A/AOmfQUcrublbtWs=">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</latexit> For p = 2: <latexit sha1_base64="2Vjt5TdfsjhzL2RS6sYja/1FGHY=">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</latexit> ||⇠||2 ||·||2 = | R xd⇠(x)|2 <latexit 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For p = 1: ˙ H d+1 2 (Rd) Sobolev norm;
  15. Sinkhorn Divergences Positivity concave <latexit 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</latexit> <latexit 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</latexit> 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</latexit>

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</latexit> concave <latexit 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bl15UpzpYSjuodqRz1SR9DGt9XbRnU8Lz1Vsks/VfaNV1xLJmWu3id8yH+NSEexNFctmj6qfYjv83edHYM8WOtLVi3ZcwfF3BdkK+wDJX+1rFVU24rnZuQUQrEOOY6v/6BUPIfE8+48DsbdmnxSJX8z3CQnn8yEFGWzS5rX+j1oO6x3V7tuEyfpN+wVva6aWtZeLkEeactobsaY08pLEW1omJzFuqq2Vpc2ijfj1bUw225lpaqurX1fLO3X3c6q5F92yP9tWhhHVvV82Z0KC/uN7fx/287beKg7b2VBN/VVv7GlX1Vbuqkfsu3IvwujSzXMN3q5zptMCrfMtfQtc3L3VmgcmOcge95TK2ZqWKmy4m9WyPL4fCzi2+N6M34muqp7J3PMGJxPHMhdm+ZO3L5egeL4rU3zDGLPc4rnptQ6+A7elpJTsB8quRu5GjkqoT7IRcdFTIu4TUmjReT72Xgyfx8p7ovGGQT79IbMHib6JBbeToeWmuSQ9zN6Yl8hukVqfHu3zN6NChrm0x3mBuEkV758Xtlry9FjkklkZJSd01eZdImn3LwvSqQS6nn+fDdyk3Ii9XsZfVs2I0GHViyrdJWQVfu0aU68rCvee96jOuYTKDOFu2pnDo7VnN6eQ5N0wvNYth+VeYOihk3+sudliuxHh+QZYiia/GWKVbNIxXJj+1jX3jmpRLm1aWOrOVbV6cKKn0OcY2iIFt10QrWFlNaiKIVqKU/J5Y0TXY8oS5daeZuskvftTWnMMNF+lncMoXbD96gvKvMd53wi3/U/ID3iboiWLnMVj2LO6nMefL5iSijzbs7gSRvbUqe0czfRs1E413vlKWWfZtumqqvXcUSz++qheqT+Bmh1IfcbbymmOo9gk9J+LP9+isPsNj97h3D4dqQN9Sy7o7FqH6G88yjlmb7nzU93J0C1THcno+unHJLXJ7Gf8kP1lMYNdbTBUp9G7Ch+RHev+SjjTXOnlKtRuiG/mu4WyPCwVPeSGtqRJHul+CZokca+DfAWnSDkexfQDqe1pGtRWhwXk/MmnA70zK3r/Jb5LqTLPa3Ljr4FyJf7Su+KDO2FvKJScM6XSvbljoP0J0rOY8oY+4rOZsuI2sQLYSr5M0PmJCWXwaYp4/YQ3atMPp9Ncx6+82yuiveJuSw9hqfcicT6yFMtajiG2m6OHtst780uShxPnWkfZCU3tPMUVzJucypXXH0eqJFFdwWQtyPtSfxb0QZiuPK5IvvziqWra7qtq0p3YfnkN4OGdHuisdYq28LbF12toY7XMDwXJRnmmZ0VrTaO4gMnTVtfC6+thXeTVsss/ZGPwhH0NXzj8VFE38U5TT+Fd4nmW0N4rCFUNujMxbGDxkmQyoOMClpAV1PK0zmJKD2PTfPnjtxnsv2nVVvZXat4++kp1cDd2rV5oOS3Gswn/N5Hz0+Rd27zmFbafMc65bumIySOb8wpKTz/cY/SXsDDVnqqz4In6q9LktjoGY08Xqo7Ksl8ir8O+E4o6UnttBWLEo6vbkfZl21dTyKtydiSP+9mlndPxd1wiAjB2LcN5CnES7sPozo3PZY+ntKRphNq+fsZv/yIakTcngTQX6iqu2a/UAtnbygWw/NF8gtA9q/zjAhhTrMw73x+/MUgTOmDJQ8o/p2per/ehBR4Jw+itumtq3f9sG8Y0ugZIzS5E8L8/hCfPseWhrNW/rNpeOp8qFyzxMXfIuF5Hz6bg1LyObNjKNOQzrp/pD3WSYR3GBCF4u04NocWzTHxzgIcOz9U/xik+op00yyczArJ/yRDJYUyhM5AoLfmswYhfnZpjqFd2tgyV7n9aQ3sZxf8Xr1TUx2tV4khwidWz3PyG19sboE41d6qFVGzVfdZx/xew0HW3st3v1TdGL0CmNCo9aDkR2waMRRmqnh/NLePWbBUcothERneddhRxTvYuV46EdbZo3Fdr9AW8hI8oLKv6lEs1vTdLKLzUX9B90+zf74I9lsvrPz4dgq9iPvMa/77mNkDiQY4ai9aqQ99rWPHFRrfuGIFP/ZSo+vFH+G4Sn7DoxgLuvoqd+zmt4uBkvPBxZucfSedZLVoO+PgnkfsKPMLm0yVz/I2KC2l2jZW2KHP+uz/6bvLa8Vfei2/PF+/v7Z6f+2z9eWf/YP+FdgfqT9Xfwm6WFMfqp+px2CTRyDTG/Uv6t/Vf1xeXv7z5b9e/htn/eEPNOZPVe6/y//6PxTZQQ0=</latexit> 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</latexit> <latexit sha1_base64="17EnyIrs9y/b+aOK904a26808CE=">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</latexit> which is, if C(x, y) = kx yk, a well-known quantity: the Energy Distance MMD. Interpolating between two standard families of divergences, the Sinkhorn divergences could be expected to share some of their desirable properties. The present paper shows that this is, indeed, what happens: we prove that S" is a convex function of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) Information theoretic Sinkhorn divergences. Before moving on to our contributions, let us mention [AKO17] for the definition of another positive divergence based on Sinkhorn’s iterations, over the discrete probability simplex. The present work opts for a rather di↵erent point of view, as we put forward the geometric structure of our feature space and handle continuous probability densities. which is, if C(x, y) = kx yk, a well-known quantity: the Energy Distance MMD. Interpolating between two standard families of divergences, the Sinkhorn divergences could be expected to share some of their desirable properties. The present paper shows that this is, indeed, what happens: we prove that S" is a convex function of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) Information theoretic Sinkhorn divergences. Before moving on to our contributions, let us mention [AKO17] for the definition of another positive divergence based on Sinkhorn’s iterations, over the discrete probability simplex. The present work opts for a rather di↵erent point of view, as we put forward the geometric structure of our feature space and handle continuous probability densities. Theorem: <latexit 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<latexit 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<latexit 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<latexit 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[Feydy, S´ ejourn´ e, P, Vialard, Trouv´ e, Amari 2018] <latexit 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<latexit 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<latexit sha1_base64="+1SAgCrta7UxoRWrAdXNR0bYA1Q=">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</latexit> If e dp " is positive: <latexit 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<latexit 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</latexit>
  16. Shrödinger and kernels dimensionality Curse of meets Shrödinger Gromov X

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convergence Sinkhorn and
  17. Sinkhorn’s Algorithm Dual problem: <latexit sha1_base64="VsKDmXK1z2Kjt/xSUsYS4msylwM=">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</latexit> <latexit 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<latexit 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<latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. = <latexit sha1_base64="1fc+j8d8XBqoBLottFct+NtS058=">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</latexit> sup (f,g)2C(X)2 Z fd↵ + Z gd + " Z X2 (1 e dp+f g " )d↵ ⌦ d
  18. Sinkhorn’s Algorithm Dual problem: <latexit sha1_base64="VsKDmXK1z2Kjt/xSUsYS4msylwM=">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</latexit> <latexit sha1_base64="VsKDmXK1z2Kjt/xSUsYS4msylwM=">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</latexit> <latexit 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    <latexit 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Soft c-transforms: <latexit 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fc,"(y) def. = min ↵ "(dp(·, y) f) <latexit 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<latexit 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gc,"(x) def. = min"(dp(x, ·) g) <latexit 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<latexit 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<latexit 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<latexit 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Soft min: <latexit 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min ↵ "(h) def. = " log Z X e h/"d↵ <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. = <latexit sha1_base64="1fc+j8d8XBqoBLottFct+NtS058=">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</latexit> sup (f,g)2C(X)2 Z fd↵ + Z gd + " Z X2 (1 e dp+f g " )d↵ ⌦ d
  19. Sinkhorn’s Algorithm Dual problem: <latexit sha1_base64="VsKDmXK1z2Kjt/xSUsYS4msylwM=">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</latexit> <latexit sha1_base64="VsKDmXK1z2Kjt/xSUsYS4msylwM=">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</latexit> <latexit 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    <latexit 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Soft c-transforms: <latexit 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fc,"(y) def. = min ↵ "(dp(·, y) f) <latexit 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<latexit 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gc,"(x) def. = min"(dp(x, ·) g) <latexit 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<latexit 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<latexit 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<latexit 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Soft min: <latexit 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min ↵ "(h) def. = " log Z X e h/"d↵ <latexit 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<latexit 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<latexit 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<latexit 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Richard Sinkhorn Only matrix/vector multiplications. ! Convolution on regular grids, separable kernels. <latexit 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<latexit 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<latexit 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Discrete measures: <latexit 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<latexit 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<latexit sha1_base64="EQZwg6x6C976cllsGR9epfwOm2Q=">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</latexit> Sinkhorn’s algorithm: <latexit 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<latexit 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<latexit sha1_base64="5QxnzlaOeMrx2y527LIbv6KfndU=">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</latexit> fk+1 def. = (gk)c," <latexit sha1_base64="+c+nAJ/SGm0FoRamb6DVcpO7CQI=">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</latexit> gk+1 def. = (fk+1)c," <latexit sha1_base64="Sq6CEtaetRw1pwC+y2D+nKwLZdI=">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</latexit> W" p (↵, )p def. = <latexit sha1_base64="1fc+j8d8XBqoBLottFct+NtS058=">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</latexit> sup (f,g)2C(X)2 Z fd↵ + Z gd + " Z X2 (1 e dp+f g " )d↵ ⌦ d
  20. Hilbert Metric Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="w40fYjAuiccyYasL2x+oGV5/rTE=">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</latexit> Dual cost:

    <latexit sha1_base64="2IxW0ycscoWUjrjOFtjQCrd9/eY=">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</latexit> W(k)(↵, ) def. = R fkd↵ + R gkd <latexit sha1_base64="MwXA7BWimeWxVVM0J3q1pg8adKw=">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</latexit> |W(k)(↵, ) W" p (↵, )p| 6 <latexit 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Fast in term of k . . . <latexit 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slow in term of " <latexit 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! useless to approximate Wp <latexit 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C(1 e | |d| | p 1 " )k
  21. Hilbert Metric Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="w40fYjAuiccyYasL2x+oGV5/rTE=">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</latexit> Dual cost:

    <latexit sha1_base64="2IxW0ycscoWUjrjOFtjQCrd9/eY=">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</latexit> W(k)(↵, ) def. = R fkd↵ + R gkd <latexit sha1_base64="MwXA7BWimeWxVVM0J3q1pg8adKw=">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</latexit> |W(k)(↵, ) W" p (↵, )p| 6 <latexit 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Fast in term of k . . . <latexit 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slow in term of " <latexit 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! useless to approximate Wp <latexit sha1_base64="cWnmvuc7+BCm+yhq1JcDkdP5n/Q=">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</latexit> Variation semi-norm: <latexit sha1_base64="eL1y4Y1OK4V7QbmiGgX0he+kONs=">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</latexit> ||h||V , sup(f) inf(f) <latexit 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||fk f ?||V = O(1 e | |dp| |1 " )k <latexit 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Birkho↵’s contraction theorem: <latexit 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f 7! fc," is contractant for || · ||V <latexit 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=) <latexit sha1_base64="CLxm7Hhiztx3QU473hecJZYnGtA=">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</latexit> Proof: <latexit sha1_base64="LXszBpp0uOZgr5r45tsOcMvo/sg=">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</latexit> C(1 e | |d| | p 1 " )k
  22. Hilbert vs Mirror Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">AABDPnictVxbcxu3FYbTW+zenPSxL5sq7jgdx5UdT9M005lYF8uKZZs2KdlJaHt4WVG0l1yaS8qyGf2Vvrb/ov0b/QOd9qGT1z72XIAFlsTuwaqudiRjQXznHBwAB+ccgO5OkmE2W1//+7l3vvf9H/zwR++ev/Djn/z0Zz+/+N77B1k6n/bi/V6apNPH3U4WJ8NxvD8bzpL48WQad0bdJH7UfbGJnz86jqfZMB23Zq8n8ZNRZzAeHg57nRlUPbv4fnsWnwBu0TqK02k8+sNp9Ozi2vrVdfqJVgvXdGFN6Z9G+t4HF1Rb9VWqemquRipWYzWDcqI6KoPnG3VNrasJ1D1RC6ibQmlIn8fqVCF2Dq1iaNGB2hfwdwBv3+jaMbwjzYzQPeCSwO8UkJG6BJgU2k2hjNwi+nxOlLG2jPaCaKJsr+HfrqY1gtqZOoJaCWdahuMyaDGivr+G3z4hq3o+U4fq99TjIWhgQjWoi57mOScdYj8jRwczoDCBOiz34fMplHuENKMSESYjTaE0Hfr8n9QSa/G9p9vO1b+oT5fgiVRT6yrNKXTUMdGPaOzn8BnLkwDnAVCItUaw9IpGZkQ6GEP7BdTfg+eUSkaDXXgWVHtaidyEx4fcFJE78PiQOyJyDx4fck9ENuDxIRsaidgp6dyPb8LjwzdFzg/g8SEfiMiH8PiQD0XkATw+5IGI/BoeH/JrEXkLHh/yloi8A48PeUdEtuDxIVsich8eH3JfRG7D40Nua2T5Sp3CkxKdobAqb0K5yAMtRQI1N0X5NsiW+rAbAWu6V4KVV/UWWVQfditAp3EJdjtg3h2WYOWZtwM20o+VbdFt2nt82NsidhdmgB+7K2K/VM9LsF8GrLQXJVh5re1BOz9Wtr534c2PvSti70HJj5X3qPtQ48feD9gxJiXYhoh9oF6WYEOs/rQEK9v9JtgVP1bep1rQ3o8NsabzEqxsTw/Ag/Fj5d3qEdT6sY9E7GN1UoJ9LGK/Auvux34VsMO+KcGaPfYC7SAD8kdiWLFV1Dr5qsTSBKh1BP5JvrdgCfeovogZ5JgBYUYiYidH7AQi9nLEXrBcWW5HM/J3ZS7NHNEMRHTzvQlLM7F9P2+PpSQAsZUjtpYQVR4pjrXpyzF5F6ZGQs7ynQtLIX1Kc/uNpVjPh2rLaxD3Cwie20c0869QtIQRFGqqitpRvsczMqL3KsQrit5MLw0PGTfLrYKLOhFRXQ+qK6Jee1CvRdTcg5qLqGMP6lhE2ZXv4toBM8DqH8diQW88A9hHLn8i8Apuwq5zG9ZoBPOnAV7gQ6q5D/82KfaWnirJMJrHfRJzIk8KlngKpYVag3obFW5RfJ3QCotBMm55X8f4+IaZkIVec2yFT/OdPMrzK+F0hiTPIKeD3mJE66kenTtUc0reHZfq4W/n696U6uG3SeOn5MVzqR5+pqWfnUH2lsa2zoBtwmqaaO3bcl0anH9hGqZ8gXZdtLg4qiM9Z5DeSU36u3pkds8wLptUYv3Ycj0amdO/rNC/OjSsnjNHz/WooPfEXq8pRbV7MtZxry3XlSGlXXSs5bBvdUemozOTC6dcj0YDPK5NirkXTrnu7J3kvbHlejQOFOc9T8mTN+V6NAb0zvqw5Xo0MNvS0XG+Lde17KgBjp1tua5VH1MWGHNAPOe5xnpFU/KT5prakPyD6myN6/Ov7mOYs3maxwjVlKxvW06nm+9l1RIZfyEGqzarKQf6F3PHByvSWKjrYnzFMswK+/sqHbvHo+b3QIsRrH4+A5By5glIaHISaL0ToHhNjLqKPTO46yIOZ8nhEqqta2eit2j5ctaoWPeMaqW4zPbW6rFN9jqjuTchn3CPNCvpYa90hMsoShraK2hIpldHd2/0ei1qf13ETZYQk3ym9ehEiM/dquNUn9abjo4v6VOeGTx85mPnL2abD7W1wZgnJVuEslTxdNuZPJJbh/vqFWVz3PxZRCOK9uqYrMaQTqQyMQo12WL2xhf0bmnv05kc8mAaPRjHSFOZKD41wyw65tMjsqiuvZV4o75Mho7LGVldY4+r0QMHPfCg68c4m7Bj3INSC2KGfXhrBUQ5F3JdpaTxqfo4P0tNaQSrI/qkYCENDbY3ccFCVkXZRwUqrwCNs4Gj9HAay3QMvr1CSY76ffLY2LVo+S/Rya05De/QHC+fzeWZmD5xvU5cI1o1fKrLb8scWIKF95Pr5L9W9xL51eGINlTi+tThzHoZ0/2AmCLYCXnGCa02aXUUW7v5qeVPDKeGMmfneJqdkoWMyP5FsD+lNCcj+nVvGpgTdLYICdnIELszzL0bn68zFOeY9eOGiu9A2PkWky2bE39D111dGc1Fjhh4HzhdmttGJ3vkC8bEdaqtu13b1bsPIu2tCneWMEU7Vy4T/4/or/k182RtZUaghnEEMm3rfOORUsyCOurQLl9tg0xbV8oPcxmeaqnt/mdl+rAg2RZFXCgP7tZ94Nyjd+aFs2RKcmcrbXgfrcrmIuXJkh6xt4cUxbPdH+gdGOW+QrvkGq25Ns2SAcyCWR5FmLZSFnmZbzWvIvUw2tn/hbrVdVFrSDFSNoPLGpLy+zFFa66UCcxqnr8vaDX5tT5dalXNZ0xzceSs5W+h9gP4a+Q272F0ugWrsEFzgCnYN6sRrolWWoTx2ijwMjPT0LLvlp+dk6aVW3OW+Jqtm42xj2tTadCsOdFZC1M+C43nDo3ngTps0Vmj1aKpN5bomRhbtPRpZSi/OtxaNSjPRcqyR2ZQwwAp3VgqjGpfpCrH+Ab1RqS1LtLqwGp1TwPcNR+C9K/15dX9bb67R+oW+TY98sA4funTKh2Sz2VqqyM1poCcb2j76q7+NtUg9y5ZUKTMtz5xxfCpU4+e01zSX+udLSU7by2Cubf0SrcxNrZN5U9WkCNaExmtS4O4QS1iLb8rR7Rkka46PkdEmf8O+VTsd1THzG5rOyZRwZ+w8SavKsuLI4Ux6V/KvO2uRK+7TvwaUUw41951F2jVH2GkwBiTSfB7lhmNEO5yfJLAHm2X7OeqneJTvLEj0VWSeqH+GGBjOOq1c92dW6bHpm+/gZaodTvqvhYyvySYo8TvLCd6HdrVRtpHXSy9n41WR+9yxfcqPcyX+Fp9zKmNG1nYKK+IaavPg7mwRPW4MCaES71e1JG/nuR1ZObTqVDKprWhXMw0sI05onhJugeKCJ93d9nrzX0k9KO7Qq9LWJca10iUMBuX6vyAa2kxK3V+ZR/i2vOVu1Hi7ERlO4Wh7u4W1n6zhYzJ+iVKytlwa1f2diFKkbMwTKGn+EZvWXzo0vwcHvwbKV90aDiG5A6b4N/eVJtq+y3chnipy5zRjKgGbUF/Kfbu6H4WW1Tr6KVD3aUfwiGcxxB0LUk/pJ20ruxMWZbcpR5O/xVZgamKRelty/p9cLnIPVnlVKc/Q7Jscm+Gynxzp25fDIeQnhS5hPPhcw2pF4fKfAOqXh8MdbkHRQ51eJh7DGFjblvX5+VyqtbXKpdQHrwLmBMXg8OTv/JYxbYLsVBTZ0TePge0DocV1M1u8b/2w/CxnOrzCuWW0XfNngeMOreLdUYW/eH6a8ZyC5nN5RzDeaZ576y35OfHfl9Ua6RSpzdvnz76o3YOGF4LxXlQWTrGu7PIyhtKBc8FfDKk6jv113PytxFe5jTK5KhDyZxTlFMzLWRq5huXvt6Zz0JksnTKZCpSs3FEk27EbqpddQt+N3MPsO7tUP4uJf+LWP+3ZvtQe0jWw2TROXPQprqYsh/2FK1P7/b+bJnEeJeX7/a2oAbPwveoFu/53qP2eNe3Vehb+TdIeK3fVanqFyKS5dM9u6660IPiyRvngMz3fCO6S89ZLL55Ngo4WzT3p5YlWtAn8s2Cbim+60jZo7k60Wf1eHKAN+w7eX4oUr+luo6287jnSpwbpZwbS5wz0k6Rw4nzWfXdrDIumw6Xfp47O9btUoqz7XledW50q5QL30Gvxg8q8ANHyiZp/wVFwlNVnc2bV9Cca5ncE9axMplI1gPGmZ18vKsj2+MKXscB/b9Tir7jSLoDsnQp/x3RCduU6CVaN9skPd90rM6k3q6Q1nyPkmnau452Hphbi9VZ+kTPO85cmP+7YEHrAXdTczNRyp7EJXS6dCMwJkp8R7Ka0qEgz6FIYSBKomcq/T8Sn9FPxIVPb+jCZ9fy/0fi4PrVa7+7+smDG2tfbOj/UeJd9Uv1K3UZ7OOn6gsYiYbaB04n6k/qz+ovG3/b+MfGvze+46bvnNOYX6jCz8Z//gv1JvJn</latexit> Theorem: <latexit sha1_base64="Fo36Govvl7SLdYwIfThuT/Rfmqo=">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</latexit> 8

    > > < > > : <latexit 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||d||2p 1 "k <latexit 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C(1 e | |d| | p 1 " )k
  23. Hilbert vs Mirror Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="Fo36Govvl7SLdYwIfThuT/Rfmqo=">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</latexit> 8

    > > < > > : <latexit 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|W(k)(↵, ) W" p (↵, )p| 6 <latexit 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(Pinsker) <latexit sha1_base64="oSRMoISiqyF5QqnMGPIkZfPFlpE=">AABB2XictVxLcxu5EYY3r7Xz8iaHHHKZROuUN+U4ktYVZ2srVStLsqw1bcsmJXvXsl18jChaQw7NIeUHV4fcUrnmJ+Sa/Ib8jvyD5JS/kH4AAwyJmcYojlGSMCC+7kYP0OhugO6Mk0E2XV3954UPvvXt73z3ex9evPT9H/zwRz++/NFPDrJ0NunG+900SSdPOu0sTgajeH86mCbxk/Ekbg87Sfy4c7KJnz8+jSfZIB21pm/H8bNhuz8aHA267Sk0vbj8s6cbyTTrHs+SeBLF06idROurazefvbi8snp9lf5Fy5U1XVlR+t9e+lG0rw5VT6Wqq2ZqqGI1UlOoJ6qtMihP1ZpaVWNoe6bm0DaB2oA+j9WZugTYGfSKoUcbWk/gdx+enurWETwjzYzQXeCSwM8EkJG6ApgU+k2gjtwi+nxGlLG1jPacaKJsb+FvR9MaQutUHUOrhDM9Q3E4lqk6Ur+nMQxgTGNqwdF1NZUZaQUlj5xRTYHCGNqw3oPPJ1DvEtLoOSJMRmNH3bbp839RT2zF567uO1P/JimvQIlUU48+zSm01SnRj+htzuAzlicBzn2gEOsxYu016XpIox9B/zm034dyRjWjkw6UObWeVSI3ofiQmyJyB4oPuSMiG1B8yIaI3IPiQ+5pJGInpHM/vgnFh2+KnB9C8SEfishHUHzIRyLyAIoPeSAiv4biQ34tIm9D8SFvi8i7UHzIuyKyBcWHbInIfSg+5L6I3IbiQ25rZPlKnUBJic5AWJUbUC/yQEuRQMuGKN8tso4+7K2ANd0twcqregv++rFbATqNS7DbAfPuqAQrz7wdsJF+rGyL7tBu4sPeEbG7MAP82F0R+6V6WYL9MmClnZRg5bXWgH5+rGx978GTH3tPxN6Hmh8r71EPoMWPfRCwY4xLsHsi9qF6VYINsfqTEqxs95tgV/xYeZ9qQX8/NsSazkqwsj09AA/Gj5V3q8fQ6sc+FrFP1JsS7BMR+xVYdz/2q4Ad9l0J1uyxl2gH6ZM/EsOKraLWzlcl1sZArS3wT/K9JSHfuAPtEqafY/qEGYqInRyxE4ho5IhGsFxZbkcz8ndlLs0c0QxEdPK9CWtTsX8v74+1JACxlSO2FhBVHim+azOWU/IuTIuEnOY7F9ZCxpTm9htrsZ4P1ZbXIB4UEDy3j2nmX6NoCSMo1FQVteN8j2dkRM9ViNcUvZlRGh4ybppbBRf1RkR1PKiOiHrrQb0VUTMPaiaiTj2oUxFlV76LOwyYAVb/+C7m9MQzgH3k8hKBV7ABu84dWKMRzJ898AIfUcsD+Nuk2FsqVZJhNI/7JGY5nhUs8QRqc7UC7TYq3KL4OqEVFoNk3POBjvHxCXMbc73m2Aqf5Tt5lGdMwukMSJ5+Tge9xYjWUz06d6nljLw7rtXD38nXvanVw2+Txs/Ii+daPfxUSz89h+wtjW2dA9uE1TTW2rf1ujQ4/8I0TP0S7bpocfGtDvWcQXpvatLf1W9m9xzvZZNqrB9br0cjc8aXFcZXh4bVc+bouR4V9J7Y6zW1qPZIRjrutfW6MqS0i460HPap7pvBPj39Zky9Ho098Lg2KeaeO/W6s3ecj8bW69E4UJz3PCNP3tTr0ejTM+vD1uvRwGxLW8f5tl7XsqMGOHa29bpWfURZYMwB8ZznFusVTchPmmlqA/IPqrM1rs+/vI9hzuZ5HiNUU7K+bTmdTr6XVUtk/IUYrNq0phzoX8wcH6xIY67WxfiKZZgW9vdlOnaPR803QIsRrH4+A5By5glIaHISaL0ToLgmRl3FkRncuojDWXK0gDrUrVPRW7R8OWtUbHtBrVJcZkdr9XhI9jqjuTcmn7BBmpX00Ch9w2UUJQ01ChqS6dXR3Tu9XovaXxVx4wXEOJ9pXToR4pO06jjVp/Wmo+Mr+pRnCoXPfOz8xWzzkbY2GPOkZItQliqebj+TR3LbcF+9pmyOmz+L6I2ivTolqzGgE6lMjEJNtpi98Tk9W9r7dCaHPJhGF95jpKmMFZ+aYRYd8+kRWVTX3kq8UV8mQ8f1jKyuscfV6L6D7nvQ9WOcTdgx7kOtBTHDPjy1AqKcS7muUtL4RP0mPx1N6Q1WR/RJwUIaGmxv4oKFrIqyjwtUXgMaZwNH6eE0FukY/OESJTnq98ljY9ei5b9CJ7fmfLtNc7x8NpdnYnrEdZ24RrRq+FSXnxY5sARz7yfr5L9WjxL51eGINlTi+tzhzHoZ0Yl/TBHsmDzjhFabtDqKvd381OInhtOeMmfneJqdkoWMyP5FsD+lNCcj+nHvDpgTdLYICdnIELszyL0bn68zEOeY9eMGim812PkWky2bEX9D111dGc1Fjhh4HzhbmNtGJw3yBWPiOtHW3a7t6t0HkfaehDtLmKKdK1eJ/yf02/yYebKyNCNQw/gGMm3rfO8jpZgFddSmXb7aBpm+rpQf5zI811Lb/c/K9HFBsi2KuFAe3K17wLlLz8wLZ8mE5M6W+vA+WpXNRcrjBT3iaI8oime739c7MMp9jXbJFVpzhzRL+jALpnkUYfpKWeRFvtW8itTDaGf/F+pW10WtIcVI2Qwua0jK78cUrblSJjCref6e0Grya32y0Kuaz4jm4tBZy99A6y/gt5HbPIfR6RSswi2aA0zBPlmNcEu01COM160CLzMzDS37bPnZOWl6uS3nia/ZutkY+7Q2lT2aNW901sLUz0PjpUPjZaAOW3TWaLVo2o0leiHGFi19WhnKrw63Vg3KM5Gy7JEZ1CBASjeWCqPaE6nKMb5BvRNprYq02rBa3dMAd82HIP1rfXF1f5Pv7pG6Tb5Nlzwwjl96tEoH5HOZ1upIjSkg5xvavrqr/5BakHuHLChS5nucuGL41KlL5SyX9Fd6Z0vJzluLYO4tvdZ9jI09pPqnS8ghrYmM1qVB3KAesZbflSNasEjXHZ8josx/m3wq9juqY2a3t30nUcGfsPEmryrLiyOFEelfyrztLkWvu078GlFMONPedQdo1X/DSIExJpPg9ywzekO4y/FJAnu0HbKfy3aKT/FGjkTXSeq5+kOAjeGo1851d26ZEZux/Rp6otbtW/f1kPklwRwlfuc50WvTrjbUPup84fl8tNp6lys+V+lhtsDX6mNGfdzIwkZ5Rcyh+jyYC0tUjwtjQrjUG0Ud+etJXkdmPp0KpWx6G8rFTAPbmGOKl6R7oIjweXdXvd7cJ8I4Okv0OoR1qXGLRAmzcanOD7iWFrNSF5f2IW69WLkbJc5OVLZTGOrubmHtN1vImKxfoqScDfd2ZT8sRClyFoYpdBXf6C2LD12an0PB35HyRYeGY0jusAn+7YbaVNvv4TbEK13njGZELWgLeguxd1uPs9ijWkevHOou/RAO4TwGoGtJ+gHtpHVlZ8qy5C71cPqvyQpMVCxKb3vWH4PLRR7JMqc64xmQZZNHM1Dmuzh1x2I4hIykyCWcD59rSKM4UuY7TfXGYKjLIyhyqMPD3GMIe+e2d31eLqdqfS1zCeXBu4A5cTE4PPkrj1VsvxALNXHeyPvngNbhqIK62S3+13EYPpZTfV6h3DL6rtnLgLfO/WKdkUV/uP6asdxCZnM5x3CeaT466y35+bHfF9V6U6kzmvdPH/1ROwcMr7niPKgsHePdWWTlDaWC5wI+GVL1H/WPC/K3EV7lNMrkqEPJnFOUUzM9ZGrmG5e+0ZnPQmSydMpkKlKzcUSTbsRuql11G342cw+w7u1Q/i4l/0Ws//uzPWg9IuthsuicOTiktpiyH/YUrUfP9v5smcR4l5fv9ragBc/CG9SK93zvU3+869sqjK38GyS81u+pVPUKEcni6Z5dVx0YQfHkjXNA5nu+Ed2l5ywW3zwbBpwtmvtTixLN6RP5ZkGnFN9xpOzSXB3rs3o8OcAb9u08PxSp31JbW9t53HMlznulnPcWOGeknSKHN85n1XezyrhsOlx6ee7sVPdLKc6253nVudGtUi58B70a36/A9x0pm6T9E4qEJ6o6mzeroDnTMrknrCNlMpGsB4wz2/n7ro5sTyt4nQaM/24p+q4j6Q7I0qH8d0QnbBOil2jdbJP0fNOxOpN6p0Ja/T1K+t8NPqN/EVdu3tCVz9by/93gYP362u+uf/pwfeWLW/r/OfhQ/Vz9Ul2FNX5TfQHU9tQ+WdO/qr+pv2883fjjxp82/sxdP7igMT9VhX8bf/kv6a+pZg==</latexit> [Altschuler et al 2017] <latexit sha1_base64="AF5/WEibbrBBLSULZXd5jxaC8nM=">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</latexit> k def. = Wp ",p (↵, ) W(k)(↵, ) <latexit sha1_base64="t0Q52zQfZWM1Bndx3LadXDLljfk=">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</latexit> k k+1 = "KL(↵|(⇡k)1) + "KL( |(⇡k)2) <latexit 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> "||↵ (⇡k)1 ||2 1 + "|| (⇡k)2 ||2 1 <latexit 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||d||2p 1 "k <latexit 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C(1 e | |d| | p 1 " )k
  24. Hilbert vs Mirror Analysis <latexit sha1_base64="HgbzUxOjfBog8xcC8c+lRFgA1iQ=">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</latexit> Theorem: <latexit sha1_base64="Fo36Govvl7SLdYwIfThuT/Rfmqo=">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</latexit> 8

    > > < > > : <latexit 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|W(k)(↵, ) W" p (↵, )p| 6 <latexit 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(Pinsker) <latexit sha1_base64="oSRMoISiqyF5QqnMGPIkZfPFlpE=">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</latexit> [Altschuler et al 2017] <latexit sha1_base64="AF5/WEibbrBBLSULZXd5jxaC8nM=">AABDeXictVxfcxPJER8u/w7yj0se87LEkDJXHDGGyuXqKlWHZWN8CBBINtydwLWSVkJ4pRVayRh0+lT5APkc95q8JE/5AHlJ/5nZmZVmd2Yd4i3Zs7Pz6+7pnenp7hm5M4mH6Wxr64cLH/3oxz/56c8+vnjp57/45a9+ffmT3xylyXzajQ67SZxMn3fCNIqH4+hwNpzF0fPJNApHnTh61jmp4fNnp9E0HSbj1uzdJHoxCgfjYX/YDWdQdXz54dX2bnR8ErSjN72oH7Rn0dls8Wx5vGhHk3QYJ+Mbk+XLyWY7jGs32p2odj34LGv0crF5cn1pPLt6fHlj6+YW/QTrhVuysCHkTyP55Mol0RY9kYiumIuRiMRYzKAci1CkcH0nboktMYG6F2IBdVMoDel5JJYCsXNoFUGLEGpP4PcA7r6TtWO4R5opobvAJYbPFJCBuAaYBNpNoYzcAno+J8pYW0R7QTRRtnfwtyNpjaB2Jl5BrQunWvrjUmgxor6/g0+PkGU9n4m++DP1eAgamFAN6qIrec5Jh9jPwNDBDChMoA7LPXg+hXKXkOqtBIRJSVMoTUjP/0ktsRbvu7LtXPyL+nQNrkA0pa6SjEIoTol+QO9+Ds9Ynhg4D4BCJDWCpbf0ZkakgzG0X0D9I7iWVFIa7MC1oNplKbIGlw1ZcyL34bIh953IOlw2ZN2JbMBlQzYkErFT0rkd34TLhm86OT+By4Z84kQ+hcuGfOpEHsFlQx45kd/CZUN+60Teg8uGvOdEPoDLhnzgRLbgsiFbTuQhXDbkoRO5B5cNuSeRxTN1CldCdIaOWXkXynkeaCliqLnrlG+HbKkNu+Mxp7sFWPes3iWLasPueug0KsDueYy7fgHWPfL2wUbasW5bdJ/WHhv2vhN7ACPAjj1wYr8WrwuwX3vMtJMCrHuu1aGdHeu2vg/hzo596MQ+gpId616jHkONHfvYY8WYFGAbTuwT8aYA62P1pwVYt91vgl2xY93rVAva27E+1nRegHXb0yPwYOxY92r1DGrt2GdO7HNxVoB97sR+A9bdjv3GY4V9X4BVa+wlWkEG5I9EMGPLqIXZrMTSBKiFDv5xtrZgCdeonhMzyDADwoyciP0Mse+JqGeIurdcaWZHU/J33VyaGaLpiehkaxOWZs72vaw9lmIPxG6G2F1BlHmk+K5VX07Ju1A1LuQsW7mw5NOnJLPfWIrkeCi3vArxOIfgsf2KRv4NipYwgkJNlVF7la3xjAzovgzxlqI31UvFw42bZVbBRJ05UR0LquNEvbOg3jlRcwtq7kSdWlCnTpSe+Sau7TECtP7xXSzojkcA+8jFVwBewV1Yde7DHA1g/DTAC3xKNY/hb5Nib9dVJhlG87hOYk7kRc4ST6G0EBtQr6PCXYqvY5phEUjGLR/LGB/vMBOykHOOrfAyW8mDLL/iT2dI8gwyOugtBjSfqtF5QDVL8u64VA1/P5v3qlQNv0caX5IXz6Vq+JmUfnYO2VsS2zoHtgmzaSK1r8tVaXD+hWmo8iVaddHi4lsdyTGD9M4q0j+Qb+bgHO+lRiXWjy5Xo5Ea/Utz/atCQ+s5NfRcjQp6T+z1qlJQuSdjGffqclUZElpFx1IOfVf1zYQyM7kwytVoNMDjqlHMvTDKVUfvJOuNLlejcSQ477kkT16Vq9EY0D3rQ5er0cBsSyjjfF2uatlRAxw763JVqz6mLDDmgHjMc432iqbkJ80ltSH5B+XZGtPnX1/HMGfzMosRyilp37aYTidby8olUv5CBFZtVlEO9C/mhg+Wp7EQ2874imWY5db3dTp6jUfN10GLAcx+3gNw5cxjkFDlJNB6x0DxljPqyvdM4badOBwl/RVUW9bOnN6i5stZo3zdMdW64jLdW63HNtnrlMbehHzCOmnWpYd64RsuoujSUD2nITe9Krp7L+drXvtbTtxkBTHJRlqXdoR43608TrVpvWno+Jrc5ZnBxXs+evxitrkvrQ3GPAnZIpSljKfZTuWRzDpcV28InePmZwG9UbRXp2Q1hrQjlTqjUJUtZm98Qfea9iHtySEPptGF9xhIKhPBu2aYRcd8ekAW1bS3Lt6oL5Wh43JKVlfZ43L0wEAPLOjqMU4NVoxHUGpBzHAIdy2PKOdSpquEND4Vn2V7qQm9wfKIPs5ZSEWD7U2Us5BlUfarHJW3gMbRwFG6P41VOgrfXqPkjvpt8ujYNW/5r9HOrdoND2mMF4/m4kxMj7huE9eAZg3v6vLdKgeWYGF9sk3+a3kvkV8VjmhDXVxfGpxZL2M6HxBRBDshzzim2eaaHfnWZn5q9Yni1BBq7xx3sxOykAHZvwDWp4TGZEAf86SB2kFnixCTjfSxO8PMu7H5OkPnGNN+3FDwGQg93iKyZXPir+iasyulscgRA68Dy5WxrXRSJ18wIq5Tad313C5ffRCpT1WYo4Qp6rGySfyv02/1UeNkY21EoIbxDaTS1tneR0IxC+oopFW+3AaptqaUVzMZXkqp9fqnZbqak2yXIi6UB1frHnDu0j3zwlEyJbnTtTa8jpZlc5HyZEWP2Ns+RfFs9wdyBUa5b9AquUFzrk2jZACjYJZFEaqtK4u8yrecV566H+30/0Jd6zqvNaQYCJ3BZQ258vsRRWumlDGMah6/JzSb7FqfrrQq5zOmsTgy5vL3UHsFfiu51b0fnU7OKuzQGGAK+k5rhGuCtRZ+vHZyvNTIVLT0veanx6RqZdacJ75m66Zj7NPKVBo0as5k1kKVz0PjtUHjtacOW7TXqLWo6pUlOnbGFi25W+nLrwq3VgXKcydlt0emUEMPKc1Yyo9qz0nVHeMr1HsnrS0nrRBmq7kbYM55H6R9rq/O7u+z1T0Q98i36ZIHxvFLj2bpkHwuVVseqTEF5HxH2ldz9repBrl3yIIiZT71iTOGd526dC0zSf8gV7aE7Ly2COrc0lvZRtnYNpVvryFHNCdSmpcKcYdaRFJ+U45gxSLdNHyOgDL/IflU7HeUx8xma/1Ogpw/oeNNnlWaF0cKY9K/K/N2sBa9Hhjxa0Ax4Vx61x2gVf0NIwXGqEyC3bNM6Q3hKsc7CezRdsh+rtsp3sUbGxLdJKkX4i8eNoajXj3WzbGleqz69im0RK3rt25r4eYXe3N08TvPjl5Iq9pI+qiLlfvz0QrlKpe/L9PDfIWv1sec2piRhY7y8pi2+NKbC0tUjQtjfLhU60UV+atJXkVm3p3ypaxaK8r5TAPbmFcUL7nOgSLC5t1tWr25645+dNbodQhrUuMaFyXMxiUyP2BaWsxKXVxbh7j2YulqFBsrUdFKoaibq4W232whI7J+sXDlbLi1KXs7F6W4szBMoSv4RG9RfGjS/BIu/B0IW3SoOPrkDpvg394VNbH3AU5DvJFlzmgGVIO2oLcSe4eyn/kW5Tp6Y1A36ftw8OcxBF27pB/SSlpVdqbsltyk7k//LVmBqYic0uuW1ftgcnH3ZJ1Tlf4MybK5ezMU6ps7VfuiOPj0JM/Fnw/va7h60RfqG1DV+qCou3uQ51CFhzrH4PfOdevqvExO5fpa5+LLg1cBteOicLjzVxyr6HY+FmpqvJEPzwGtQ7+Eulot/td+KD6aU3VevtxS+q7Za4+3zu0imZFFf7j6nNHcfEZzMUd/nknWO+0t2fmx3xdUelOJ0ZsPTx/9UT0GFK+F4DyoWzrGm6NIy+tLBfcFbDIk4t/ibxfc30Z4k9EokqMKJbVPUUxNtXBTU9+4tPVOPfORSdMpkilPTccRTToRWxMH4h58apkHWPV0KH+Xkv8i1v6t2R7U9sl6qCw6Zw7aVBdR9kPvovXoXp+fLZIYz/Ly2d4W1OBeeJ1q8ZzvI2qPZ31bub4Vf4OE5/pDkYheLiJZ3d3T86oDPcjvvHEOSH3PN6Cz9JzF4pNnI4+9RXV+alWiBT1xnyzoFOI7hpRdGqsTuVePOwd4wj7M8kOB+CPVhdLO45rr4two5NxY4ZySdvIczoxn5WezirjUDC69LHd2KtslFGfr/bzy3OhuIRc+g16OH5TgB4aUTdL+CUXCU1GezZuX0JxLmcwd1rFQmUjWA8aZYfa+yyPb0xJepx79f1CIfmBIug+ydCj/HdAO25ToxVI3eyQ9n3Qsz6TeL5FWfY+SaeqzjnocqFOL5Vn6WI47zlyo/12woPmAq6k6mejKnkQFdDp0IjAiSnxGspxS3yFP30lh4JREjlT6PxJf0E/Ahc/vyMIXt7L/I3G0ffPWn27efrK98dWO/I8SH4vfid+LTbCPn4uv4E00xCFw+qv4Qfxd/GPnP7Urtc3ap9z0owsS81uR+6nd/i/0+wRu</latexit> k def. = Wp ",p (↵, ) W(k)(↵, ) <latexit sha1_base64="t0Q52zQfZWM1Bndx3LadXDLljfk=">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</latexit> k k+1 = "KL(↵|(⇡k)1) + "KL( |(⇡k)2) <latexit 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> "||↵ (⇡k)1 ||2 1 + "|| (⇡k)2 ||2 1 <latexit 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||d||2p 1 "k <latexit 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Proposition: <latexit sha1_base64="khAXsjYAZa9k9xN+ghEAuxw2Rhk=">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</latexit> for ↵ = 1 n P i xi <latexit 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" = log(n) <latexit 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operations <latexit 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|W(k) Wp p | 6 <latexit 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in n2 log(n)||d||2p 1 /"2 <latexit 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C(1 e | |d| | p 1 " )k
  25. Shrödinger and kernels inkhorn divergence. In many imaging and ML

    applica- registration or GAN training), ↵ is a parameterized distribu- model to an observed empirical measure , the entropic bias imental to the accuracy of the whole pipeline. Recently, with ts, [GPC18] and [SZRM18] (see also [SBRL18]) thus proposed d regularized OT cost, or Sinkhorn divergence, ) def. = OT" (↵, ) 1 2 OT" (↵, ↵) 1 2 OT" ( , ) (3) y least, S" ( , ) = 0. The insight shared by these papers is nverges to an average cost h↵, C ? i when " tends to infinity that " (↵, ) "!+1 ! 1 2 h ↵ , C ? (↵ ) i; (4) y) = kx yk, a well-known quantity: the Energy Distance ng between two standard families of divergences, the Sinkhorn be expected to share some of their desirable properties. aper shows that this is, indeed, what happens: we prove that ction of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) eoretic Sinkhorn divergences. Before moving on to our us mention [AKO17] for the definition of another positive on Sinkhorn’s iterations, over the discrete probability simplex. opts for a rather di↵erent point of view, as we put forward cture of our feature space and handle continuous probability heuristic arguments, [GPC18] and [SZRM18] (see also [SBRL18]) thus proposed to use an unbiased regularized OT cost, or Sinkhorn divergence, S" (↵, ) def. = OT" (↵, ) 1 2 OT" (↵, ↵) 1 2 OT" ( , ) (3) so that at the very least, S" ( , ) = 0. The insight shared by these papers is that since OT" converges to an average cost h↵, C ? i when " tends to infinity (1), we can show that 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. (a) Regularized OT, OT"(↵, ). (b) Sinkhorn divergence, S"(↵, ). Figure 2: Removing the entropic bias. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are dimensionality Curse of meets Shrödinger Gromov X <latexit 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convergence Sinkhorn and
  26. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit 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Theorem: E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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  27. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit 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Theorem: E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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  28. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit sha1_base64="iqmR2w6gptZhqD4tCYOzKsFQvt8=">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</latexit> Theorem: E(|||ˆ ↵ ˆ||k ||↵ ||k |) = O(n 1 2 ) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">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</latexit> E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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<latexit 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  29. Sample Complexity n ↵ <latexit 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<latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit sha1_base64="iqmR2w6gptZhqD4tCYOzKsFQvt8=">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</latexit> Theorem: E(|||ˆ ↵ ˆ||k ||↵ ||k |) = O(n 1 2 ) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">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</latexit> E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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  30. Leveraging smoothness in high dimension? <latexit sha1_base64="ePzgW+JqdbDqdwOItscJ4SbXEx4=">AABB1XictVzdchS5FRabv4X8sdnL3HRiSEGKJcZLZbO1lao1tjFeBmyYsWF3DdT8tIeBnulhesbYzPoulds8Qm6Tp8hz5A2Sq7xCzo/UUs+o+6gdgsq2WqPvnKPT0tE5Rxo642SQTVdX/3nhg+99/wc//NGHFy/9+Cc//dnPL3/0i4MsnU268X43TdLJ0047i5PBKN6fDqZJ/HQ8idvDThI/6bzewM+fHMeTbJCOWtPTcfxs2O6PBkeDbnsKTS8uf7yRDsdJfDKYnkZXdq+Nnq9dv/Li8srqzVX6Fy1XbunKitL/9tKPon11qHoqVV01U0MVq5GaQj1RbZVB+VbdUqtqDG3P1BzaJlAb0OexOlOXADuDXjH0aEPra/jdh6dvdesInpFmRugucEngZwLISF0FTAr9JlBHbhF9PiPK2FpGe040UbZT+NvRtIbQOlUvoVXCmZ6hOBzLVB2pP9AYBjCmMbXg6Lqayoy0gpJHzqimQGEMbVjvwecTqHcJafQcESajsaNu2/T5v6gntuJzV/edqX+TlFehRKqpR5/mFNrqmOhH9DZn8BnLkwDnPlCI9Rix9pZ0PaTRj6D/HNofQjmjmtFJB8qcWs8qkRtQfMgNEbkNxYfcFpENKD5kQ0TuQfEh9zQSsRPSuR/fhOLDN0XOj6D4kI9E5GMoPuRjEXkAxYc8EJHfQPEhvxGRd6H4kHdF5H0oPuR9EdmC4kO2ROQ+FB9yX0RuQfEhtzSyfKVOoKREZyCsynWoF3mgpUigZV2U7w5ZRx/2TsCa7pZg5VW9CX/92M0AncYl2K2AeXdUgpVn3jbYSD9WtkX3aDfxYe+J2B2YAX7sjoj9Sr0qwX4VsNJel2DltdaAfn6sbH0fwJMf+0DEPoSaHyvvUbvQ4sfuBuwY4xLsnoh9pN6UYEOs/qQEK9v9JtgVP1bep1rQ348NsaazEqxsTw/Ag/Fj5d3qCbT6sU9E7FN1UoJ9KmK/Buvux34dsMO+K8GaPfYS7SB98kdiWLFV1Nr5qsTaGKi1Bf5Jvrck5Bt3oF3C9HNMnzBDEbGdI7YDEY0c0QiWK8vtaEb+rsylmSOagYhOvjdhbSr27+X9sZYEIDZzxOYCosojxXdtxnJM3oVpkZDTfOfCWsiY0tx+Yy3W86Ha8hrEbgHBc/slzfwbFC1hBIWaqqL2Mt/jGRnRcxXiLUVvZpSGh4yb5lbBRZ2IqI4H1RFRpx7UqYiaeVAzEXXsQR2LKLvyXdxhwAyw+sd3MacnngHsI5eXCLyCddh17sEajWD+7IEX+JhaduFvk2JvqVRJhtE87pOY5XhWsMQTqM3VCrTbqHCT4uuEVlgMknHPXR3j4xPmNuZ6zbEVPst38ijPmITTGZA8/ZwOeosRrad6dO5Tyxl5d1yrh7+Xr3tTq4ffIo2fkRfPtXr4qZZ+eg7ZWxrbOge2CatprLVv63VpcP6FaZj6Jdp10eLiWx3qOYP0TmrS39FvZucc72WDaqwfW69HI3PGlxXGV4eG1XPm6LkeFfSe2Os1taj2SEY67rX1ujKktIuOtBz2qe6bwT49/WZMvR6NPfC4Nijmnjv1urN3nI/G1uvROFCc9zwjT97U69Ho0zPrw9br0cBsS1vH+bZe17KjBjh2tvW6Vn1EWWDMAfGc5xbrFU3IT5ppagPyD6qzNa7Pv7yPYc7meR4jVFOyvm05nU6+l1VLZPyFGKzatKYc6F/MHB+sSGOu1sT4imWYFvb3ZTp2j0fNN0CLEax+PgOQcuYJSGhyEmi9E6B4S4y6iiMzuDURh7PkaAF1qFunordo+XLWqNj2glqluMyO1urxkOx1RnNvTD5hgzQr6aFR+obLKEoaahQ0JNOro7t3er0Wtb8q4sYLiHE+07p0IsQnadVxqk/rTUfHV/UpzxQKn/nY+YvZ5iNtbTDmSckWoSxVPN1+Jo/ktuG+ekPZHDd/FtEbRXt1TFZjQCdSmRiFmmwxe+Nzera09+lMDnkwjS68x0hTGSs+NcMsOubTI7Korr2VeKO+TIaO6xlZXWOPq9F9B933oOvHOBuwYzyEWgtihn14agVEOZdyXaWk8Yn6JD8dTekNVkf0ScFCGhpsb+KChayKsl8WqLwFNM4GjtLDaSzSMfjDJUpy1O+Tx8auRct/lU5uzfl2m+Z4+Wwuz8T0iOsacY1o1fCpLj8tcmAJ5t5P1sh/rR4l8qvDEW2oxPW5w5n1MqIT/5gi2DF5xgmtNml1FHu7+anFTwynPWXOzvE0OyULGZH9i2B/SmlORvTj3h0wJ+hsERKykSF2Z5B7Nz5fZyDOMevHDRTfarDzLSZbNiP+hq67ujKaixwx8D5wtjC3jU4a5AvGxHWirbtd29W7DyLtPQl3ljBFO1euEf/r9Nv8mHmysjQjUMP4BjJt63zvI6WYBXXUpl2+2gaZvq6UV3IZnmup7f5nZbpSkGyTIi6UB3frHnDu0jPzwlkyIbmzpT68j1Zlc5HyeEGPONojiuLZ7vf1Doxy36BdcoXW3CHNkj7MgmkeRZi+UhZ5kW81ryL1MNrZ/4W61XVRa0gxUjaDyxqS8vsxRWuulAnMap6/r2k1+bU+WehVzWdEc3HorOXvoPVX8NvIbZ7D6HQKVuEOzQGmYJ+sRrglWuoRxutOgZeZmYaWfbb87Jw0vdyW88TXbN1sjH1cm8oezZoTnbUw9fPQeOXQeBWowxadNVotmnZjiV6IsUVLn1aG8qvDrVWD8kykLHtkBjUIkNKNpcKo9kSqcoxvUO9EWqsirTasVvc0wF3zIUj/Wl9c3d/lu3uk7pJv0yUPjOOXHq3SAflcprU6UmMKyPm2tq/u6j+kFuTeIQuKlPkeJ64YPnXqUjnLJf2N3tlSsvPWIph7S291H2NjD6n+6RJySGsio3VpELepR6zld+WIFizSTcfniCjz3yafiv2O6pjZ7W3fSVTwJ2y8yavK8uJIYUT6lzJvO0vR644Tv0YUE860d90BWvXfMFJgjMkk+D3LjN4Q7nJ8ksAebYfs57Kd4lO8kSPRTZJ6rv4YYGM46rVz3Z1bZsRmbL+Fnqh1+9Z9PWR+STBHid95TvTatKsNtY86X3g+H6223uWKz1V6mC3wtfqYUR83srBRXhFzqL4I5sIS1ePCmBAu9UZRR/56kteRmU+nQimb3oZyMdPANuYlxUvSPVBE+Ly7a15v7rowjs4SvQ5hXWrcIlHCbFyq8wOupcWs1MWlfYhbL1buRomzE5XtFIa6u1tY+80WMibrlygpZ8O9XdkPC1GKnIVhCl3FN3rL4kOX5hdQ8HekfNGh4RiSO2yCf7uuNtTWe7gN8UbXOaMZUQvagt5C7N3W4yz2qNbRG4e6Sz+EQziPAehakn5AO2ld2ZmyLLlLPZz+W7ICExWL0tue9cfgcpFHssypzngGZNnk0QyU+S5O3bEYDiEjKXIJ58PnGtIojpT5TlO9MRjq8giKHOrwMPcYwt657V2fl8upWl/LXEJ58C5gTlwMDk/+ymMV2y/EQk2cN/L+OaB1OKqgbnaL/3Ucho/lVJ9XKLeMvmv2KuCtc79YZ2TRH66/Ziy3kNlczjGcZ5qPznpLfn7s90W13lTqjOb900d/1M4Bw2uuOA8qS8d4dxZZeUOp4LmAT4ZU/Uf944L8bYQ3OY0yOepQMucU5dRMD5ma+calb3TmsxCZLJ0ymYrUbBzRpBuxG2pH3YWfjdwDrHs7lL9LyX8R6//+bA9aj8h6mCw6Zw4OqS2m7Ic9RevRs70/WyYx3uXlu70taMGz8Aa14j3fh9Qf7/q2CmMr/wYJr/UHKlW9QkSyeLpn11UHRlA8eeMckPmeb0R36TmLxTfPhgFni+b+1KJEc/pEvlnQKcV3HCm7NFfH+qweTw7whn07zw9F6nfU1tZ2HvdcifNeKee9Bc4ZaafI4cT5rPpuVhmXDYdLL8+dHet+KcXZ9jyvOje6WcqF76BX4/sV+L4jZZO0/5oi4YmqzubNKmjOtEzuCetImUwk6wHjzHb+vqsj2+MKXscB479fir7vSLoNsnQo/x3RCduE6CVaN1skPd90rM6k3quQVn+Pkv53g8/pX8SVz27ryue38v/d4GDt5q3f3/z00drKl3f0/3Pwofql+rW6Bmv8M/UlUNtT+8DhVP1V/U39ff3J+tn6n9b/zF0/uKAxH6vCv/W//BeD66fT</latexit> Complexity O(n2)

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

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

  33. Leveraging smoothness in high dimension? <latexit sha1_base64="ePzgW+JqdbDqdwOItscJ4SbXEx4=">AABB1XictVzdchS5FRabv4X8sdnL3HRiSEGKJcZLZbO1lao1tjFeBmyYsWF3DdT8tIeBnulhesbYzPoulds8Qm6Tp8hz5A2Sq7xCzo/UUs+o+6gdgsq2WqPvnKPT0tE5Rxo642SQTVdX/3nhg+99/wc//NGHFy/9+Cc//dnPL3/0i4MsnU268X43TdLJ0047i5PBKN6fDqZJ/HQ8idvDThI/6bzewM+fHMeTbJCOWtPTcfxs2O6PBkeDbnsKTS8uf7yRDsdJfDKYnkZXdq+Nnq9dv/Li8srqzVX6Fy1XbunKitL/9tKPon11qHoqVV01U0MVq5GaQj1RbZVB+VbdUqtqDG3P1BzaJlAb0OexOlOXADuDXjH0aEPra/jdh6dvdesInpFmRugucEngZwLISF0FTAr9JlBHbhF9PiPK2FpGe040UbZT+NvRtIbQOlUvoVXCmZ6hOBzLVB2pP9AYBjCmMbXg6Lqayoy0gpJHzqimQGEMbVjvwecTqHcJafQcESajsaNu2/T5v6gntuJzV/edqX+TlFehRKqpR5/mFNrqmOhH9DZn8BnLkwDnPlCI9Rix9pZ0PaTRj6D/HNofQjmjmtFJB8qcWs8qkRtQfMgNEbkNxYfcFpENKD5kQ0TuQfEh9zQSsRPSuR/fhOLDN0XOj6D4kI9E5GMoPuRjEXkAxYc8EJHfQPEhvxGRd6H4kHdF5H0oPuR9EdmC4kO2ROQ+FB9yX0RuQfEhtzSyfKVOoKREZyCsynWoF3mgpUigZV2U7w5ZRx/2TsCa7pZg5VW9CX/92M0AncYl2K2AeXdUgpVn3jbYSD9WtkX3aDfxYe+J2B2YAX7sjoj9Sr0qwX4VsNJel2DltdaAfn6sbH0fwJMf+0DEPoSaHyvvUbvQ4sfuBuwY4xLsnoh9pN6UYEOs/qQEK9v9JtgVP1bep1rQ348NsaazEqxsTw/Ag/Fj5d3qCbT6sU9E7FN1UoJ9KmK/Buvux34dsMO+K8GaPfYS7SB98kdiWLFV1Nr5qsTaGKi1Bf5Jvrck5Bt3oF3C9HNMnzBDEbGdI7YDEY0c0QiWK8vtaEb+rsylmSOagYhOvjdhbSr27+X9sZYEIDZzxOYCosojxXdtxnJM3oVpkZDTfOfCWsiY0tx+Yy3W86Ha8hrEbgHBc/slzfwbFC1hBIWaqqL2Mt/jGRnRcxXiLUVvZpSGh4yb5lbBRZ2IqI4H1RFRpx7UqYiaeVAzEXXsQR2LKLvyXdxhwAyw+sd3MacnngHsI5eXCLyCddh17sEajWD+7IEX+JhaduFvk2JvqVRJhtE87pOY5XhWsMQTqM3VCrTbqHCT4uuEVlgMknHPXR3j4xPmNuZ6zbEVPst38ijPmITTGZA8/ZwOeosRrad6dO5Tyxl5d1yrh7+Xr3tTq4ffIo2fkRfPtXr4qZZ+eg7ZWxrbOge2CatprLVv63VpcP6FaZj6Jdp10eLiWx3qOYP0TmrS39FvZucc72WDaqwfW69HI3PGlxXGV4eG1XPm6LkeFfSe2Os1taj2SEY67rX1ujKktIuOtBz2qe6bwT49/WZMvR6NPfC4Nijmnjv1urN3nI/G1uvROFCc9zwjT97U69Ho0zPrw9br0cBsS1vH+bZe17KjBjh2tvW6Vn1EWWDMAfGc5xbrFU3IT5ppagPyD6qzNa7Pv7yPYc7meR4jVFOyvm05nU6+l1VLZPyFGKzatKYc6F/MHB+sSGOu1sT4imWYFvb3ZTp2j0fNN0CLEax+PgOQcuYJSGhyEmi9E6B4S4y6iiMzuDURh7PkaAF1qFunordo+XLWqNj2glqluMyO1urxkOx1RnNvTD5hgzQr6aFR+obLKEoaahQ0JNOro7t3er0Wtb8q4sYLiHE+07p0IsQnadVxqk/rTUfHV/UpzxQKn/nY+YvZ5iNtbTDmSckWoSxVPN1+Jo/ktuG+ekPZHDd/FtEbRXt1TFZjQCdSmRiFmmwxe+Nzera09+lMDnkwjS68x0hTGSs+NcMsOubTI7Korr2VeKO+TIaO6xlZXWOPq9F9B933oOvHOBuwYzyEWgtihn14agVEOZdyXaWk8Yn6JD8dTekNVkf0ScFCGhpsb+KChayKsl8WqLwFNM4GjtLDaSzSMfjDJUpy1O+Tx8auRct/lU5uzfl2m+Z4+Wwuz8T0iOsacY1o1fCpLj8tcmAJ5t5P1sh/rR4l8qvDEW2oxPW5w5n1MqIT/5gi2DF5xgmtNml1FHu7+anFTwynPWXOzvE0OyULGZH9i2B/SmlORvTj3h0wJ+hsERKykSF2Z5B7Nz5fZyDOMevHDRTfarDzLSZbNiP+hq67ujKaixwx8D5wtjC3jU4a5AvGxHWirbtd29W7DyLtPQl3ljBFO1euEf/r9Nv8mHmysjQjUMP4BjJt63zvI6WYBXXUpl2+2gaZvq6UV3IZnmup7f5nZbpSkGyTIi6UB3frHnDu0jPzwlkyIbmzpT68j1Zlc5HyeEGPONojiuLZ7vf1Doxy36BdcoXW3CHNkj7MgmkeRZi+UhZ5kW81ryL1MNrZ/4W61XVRa0gxUjaDyxqS8vsxRWuulAnMap6/r2k1+bU+WehVzWdEc3HorOXvoPVX8NvIbZ7D6HQKVuEOzQGmYJ+sRrglWuoRxutOgZeZmYaWfbb87Jw0vdyW88TXbN1sjH1cm8oezZoTnbUw9fPQeOXQeBWowxadNVotmnZjiV6IsUVLn1aG8qvDrVWD8kykLHtkBjUIkNKNpcKo9kSqcoxvUO9EWqsirTasVvc0wF3zIUj/Wl9c3d/lu3uk7pJv0yUPjOOXHq3SAflcprU6UmMKyPm2tq/u6j+kFuTeIQuKlPkeJ64YPnXqUjnLJf2N3tlSsvPWIph7S291H2NjD6n+6RJySGsio3VpELepR6zld+WIFizSTcfniCjz3yafiv2O6pjZ7W3fSVTwJ2y8yavK8uJIYUT6lzJvO0vR644Tv0YUE860d90BWvXfMFJgjMkk+D3LjN4Q7nJ8ksAebYfs57Kd4lO8kSPRTZJ6rv4YYGM46rVz3Z1bZsRmbL+Fnqh1+9Z9PWR+STBHid95TvTatKsNtY86X3g+H6223uWKz1V6mC3wtfqYUR83srBRXhFzqL4I5sIS1ePCmBAu9UZRR/56kteRmU+nQimb3oZyMdPANuYlxUvSPVBE+Ly7a15v7rowjs4SvQ5hXWrcIlHCbFyq8wOupcWs1MWlfYhbL1buRomzE5XtFIa6u1tY+80WMibrlygpZ8O9XdkPC1GKnIVhCl3FN3rL4kOX5hdQ8HekfNGh4RiSO2yCf7uuNtTWe7gN8UbXOaMZUQvagt5C7N3W4yz2qNbRG4e6Sz+EQziPAehakn5AO2ld2ZmyLLlLPZz+W7ICExWL0tue9cfgcpFHssypzngGZNnk0QyU+S5O3bEYDiEjKXIJ58PnGtIojpT5TlO9MRjq8giKHOrwMPcYwt657V2fl8upWl/LXEJ58C5gTlwMDk/+ymMV2y/EQk2cN/L+OaB1OKqgbnaL/3Ucho/lVJ9XKLeMvmv2KuCtc79YZ2TRH66/Ziy3kNlczjGcZ5qPznpLfn7s90W13lTqjOb900d/1M4Bw2uuOA8qS8d4dxZZeUOp4LmAT4ZU/Uf944L8bYQ3OY0yOepQMucU5dRMD5ma+calb3TmsxCZLJ0ymYrUbBzRpBuxG2pH3YWfjdwDrHs7lL9LyX8R6//+bA9aj8h6mCw6Zw4OqS2m7Ic9RevRs70/WyYx3uXlu70taMGz8Aa14j3fh9Qf7/q2CmMr/wYJr/UHKlW9QkSyeLpn11UHRlA8eeMckPmeb0R36TmLxTfPhgFni+b+1KJEc/pEvlnQKcV3HCm7NFfH+qweTw7whn07zw9F6nfU1tZ2HvdcifNeKee9Bc4ZaafI4cT5rPpuVhmXDYdLL8+dHet+KcXZ9jyvOje6WcqF76BX4/sV+L4jZZO0/5oi4YmqzubNKmjOtEzuCetImUwk6wHjzHb+vqsj2+MKXscB479fir7vSLoNsnQo/x3RCduE6CVaN1skPd90rM6k3quQVn+Pkv53g8/pX8SVz27ryue38v/d4GDt5q3f3/z00drKl3f0/3Pwofql+rW6Bmv8M/UlUNtT+8DhVP1V/U39ff3J+tn6n9b/zF0/uKAxH6vCv/W//BeD66fT</latexit> Complexity O(n2)

  34. Shrödinger and kernels dimensionality Curse of meets Shrödinger Gromov X

    <latexit 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convergence Sinkhorn and
  35. Gromov-Wasserstein ↵ 2 M1 + (X) and dX distance on

    X. <latexit 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<latexit 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<latexit 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<latexit 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Metric measure space X , (X, ↵, dX ): X <latexit 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x <latexit 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x0
  36. Gromov-Wasserstein ↵ 2 M1 + (X) and dX distance on

    X. <latexit 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<latexit 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<latexit 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<latexit 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Metric measure space X , (X, ↵, dX ): X <latexit 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dX k(x, y, x0, y0) def. = |dX (x, x0) dY (y, y0)|2 <latexit 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<latexit 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<latexit 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GW2 2 (X, Y) def. = min ⇡1=↵,⇡2= Z (X⇥Y)2 k d⇡ ⌦ ⇡ Y <latexit 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<latexit 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x0
  37. Gromov-Wasserstein ! non-convex, NP-hard . . . <latexit sha1_base64="ggcMBMVGAVaQVaHnsBSwPgZxe5I=">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</latexit> <latexit

    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if dX = || · ||, dY = || · ||: concave! ↵ 2 M1 + (X) and dX distance on X. <latexit 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<latexit 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<latexit 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Metric measure space X , (X, ↵, dX ): X <latexit 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dX k(x, y, x0, y0) def. = |dX (x, x0) dY (y, y0)|2 <latexit 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<latexit 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<latexit 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GW2 2 (X, Y) def. = min ⇡1=↵,⇡2= Z (X⇥Y)2 k d⇡ ⌦ ⇡ Y <latexit 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<latexit 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x0
  38. Gromov-Wasserstein ! non-convex, NP-hard . . . <latexit sha1_base64="ggcMBMVGAVaQVaHnsBSwPgZxe5I=">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</latexit> <latexit

    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if dX = || · ||, dY = || · ||: concave! ↵ 2 M1 + (X) and dX distance on X. <latexit 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<latexit 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<latexit 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Metric measure space X , (X, ↵, dX ): X <latexit 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[Memoli 2011] <latexit 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up to isometries. <latexit 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Theorem: GW is a distance <latexit 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[Sturm 2011] X <latexit 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dX k(x, y, x0, y0) def. = |dX (x, x0) dY (y, y0)|2 <latexit 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<latexit 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<latexit 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GW2 2 (X, Y) def. = min ⇡1=↵,⇡2= Z (X⇥Y)2 k d⇡ ⌦ ⇡ Y <latexit 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<latexit 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  39. Examples of Applications phine Adobe Research MIT Source Targets Figure

    1: Entropic GW can find correspondences between a source surface (left) and a surface with similar structure, a surface with shared semantic structure, a noisy 3D point cloud, an icon, and a hand drawing. Each fuzzy map was computed using the same code. In this paper, we propose a new correspondence algorithm that minimizes distortion of long- and short-range distances alike. We study an entropically-regularized version of the Gromov-Wasserstein (GW) mapping objective function from [M´ emoli 2011] measuring the distortion of geodesic distances. The optimizer is a probabilistic matching expressed as a “fuzzy” correspondence matrix in the style of [Kim et al. 2012; Solomon et al. 2012]; we control sharpness of the correspondence via the weight of the entropic regularizer. Although [M´ emoli 2011] and subsequent work identified the possi- bility of using GW distances for geometric correspondence, computa- tional challenges hampered their practical application. To overcome these challenges, we build upon recent methods for regularized op- 0 0.02 0.04 0 0.02 Figure 15: MDS embedding of four classes from SH 0 0.5 0 0.5 1 1 10 15 20 25 35 40 45 PCA 1 PCA 2 Figure 16: Recovery of galloping horse seq 0 is the base shape) as a feature vector for shape i. the result presented in the work of Rustamov et the circular structure of meshes from a galloping h sequence (Figure 16). Unlike Rustamov et al., howev does not require ground truth maps between shapes 5.2 Supervised Matching An important feature of a matching tool is the ability user input, e.g. ground truth matches of points or r GWα framework, one way to enforce these constrain a stencil S specifying a sparsity pattern for the map Γ constraints in this form is as simple as replacing K Algorithm 1 before Sinkhorn projection. Figure 17 illustrates a prototype “user session” in i design. Initially, we optimize with high regularizati straints, yielding a superposition of symmetric map prompted with the highest-entropy row (leftmost targ marked in black) and inputs a ground truth match, m in the remaining target images. The map is recomput regularizer and the ground truth match in S, disam rotational symmetry (center target). GWα is still u MDS in 2-D <latexit sha1_base64="SrVzka0uHt0puaznp5HP0fXKCZQ=">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</latexit> Shape registration:
  40. Examples of Applications phine Adobe Research MIT Source Targets Figure

    1: Entropic GW can find correspondences between a source surface (left) and a surface with similar structure, a surface with shared semantic structure, a noisy 3D point cloud, an icon, and a hand drawing. Each fuzzy map was computed using the same code. In this paper, we propose a new correspondence algorithm that minimizes distortion of long- and short-range distances alike. We study an entropically-regularized version of the Gromov-Wasserstein (GW) mapping objective function from [M´ emoli 2011] measuring the distortion of geodesic distances. The optimizer is a probabilistic matching expressed as a “fuzzy” correspondence matrix in the style of [Kim et al. 2012; Solomon et al. 2012]; we control sharpness of the correspondence via the weight of the entropic regularizer. Although [M´ emoli 2011] and subsequent work identified the possi- bility of using GW distances for geometric correspondence, computa- tional challenges hampered their practical application. To overcome these challenges, we build upon recent methods for regularized op- 0 0.02 0.04 0 0.02 Figure 15: MDS embedding of four classes from SH 0 0.5 0 0.5 1 1 10 15 20 25 35 40 45 PCA 1 PCA 2 Figure 16: Recovery of galloping horse seq 0 is the base shape) as a feature vector for shape i. the result presented in the work of Rustamov et the circular structure of meshes from a galloping h sequence (Figure 16). Unlike Rustamov et al., howev does not require ground truth maps between shapes 5.2 Supervised Matching An important feature of a matching tool is the ability user input, e.g. ground truth matches of points or r GWα framework, one way to enforce these constrain a stencil S specifying a sparsity pattern for the map Γ constraints in this form is as simple as replacing K Algorithm 1 before Sinkhorn projection. Figure 17 illustrates a prototype “user session” in i design. Initially, we optimize with high regularizati straints, yielding a superposition of symmetric map prompted with the highest-entropy row (leftmost targ marked in black) and inputs a ground truth match, m in the remaining target images. The map is recomput regularizer and the ground truth match in S, disam rotational symmetry (center target). GWα is still u MDS in 2-D <latexit 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Shape registration: [Rupp et al 2012] 2 6 6 6 6 4 3 7 7 7 7 5 <latexit 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Quantum chemistry: <latexit 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dX (xi, x0 i ) <latexit 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dX
  41. Examples of Applications phine Adobe Research MIT Source Targets Figure

    1: Entropic GW can find correspondences between a source surface (left) and a surface with similar structure, a surface with shared semantic structure, a noisy 3D point cloud, an icon, and a hand drawing. Each fuzzy map was computed using the same code. In this paper, we propose a new correspondence algorithm that minimizes distortion of long- and short-range distances alike. We study an entropically-regularized version of the Gromov-Wasserstein (GW) mapping objective function from [M´ emoli 2011] measuring the distortion of geodesic distances. The optimizer is a probabilistic matching expressed as a “fuzzy” correspondence matrix in the style of [Kim et al. 2012; Solomon et al. 2012]; we control sharpness of the correspondence via the weight of the entropic regularizer. Although [M´ emoli 2011] and subsequent work identified the possi- bility of using GW distances for geometric correspondence, computa- tional challenges hampered their practical application. To overcome these challenges, we build upon recent methods for regularized op- 0 0.02 0.04 0 0.02 Figure 15: MDS embedding of four classes from SH 0 0.5 0 0.5 1 1 10 15 20 25 35 40 45 PCA 1 PCA 2 Figure 16: Recovery of galloping horse seq 0 is the base shape) as a feature vector for shape i. the result presented in the work of Rustamov et the circular structure of meshes from a galloping h sequence (Figure 16). Unlike Rustamov et al., howev does not require ground truth maps between shapes 5.2 Supervised Matching An important feature of a matching tool is the ability user input, e.g. ground truth matches of points or r GWα framework, one way to enforce these constrain a stencil S specifying a sparsity pattern for the map Γ constraints in this form is as simple as replacing K Algorithm 1 before Sinkhorn projection. Figure 17 illustrates a prototype “user session” in i design. Initially, we optimize with high regularizati straints, yielding a superposition of symmetric map prompted with the highest-entropy row (leftmost targ marked in black) and inputs a ground truth match, m in the remaining target images. The map is recomput regularizer and the ground truth match in S, disam rotational symmetry (center target). GWα is still u MDS in 2-D <latexit 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Shape registration: <latexit 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! cross-domain adaptation, GW-VAE, etc. [Rupp et al 2012] 2 6 6 6 6 4 3 7 7 7 7 5 <latexit sha1_base64="mMhwcqbZoRKkmeq8+mJQjlxuDbg=">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</latexit> Quantum chemistry: <latexit 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dX (xi, x0 i ) <latexit 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dX <latexit 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[Demetci et al 2020] <latexit 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Single-cell <latexit 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multi-omics: <latexit 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chromatin <latexit sha1_base64="BRmt+3a8xccUaT6fKM1QbAAzzkg=">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</latexit> RNA expression
  42. Schrodinger GW <latexit sha1_base64="u3OSfARDH0CU0Lzklru2NWu32+o=">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</latexit> min ⇡1=↵,⇡2= Z (X⇥Y)2 k d⇡

    ⌦ ⇡ + " KL(⇡|↵ ⌦ ) <latexit sha1_base64="EES47+cAzeEuV1aOat8Q1+AdkiY=">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</latexit> d(x, y)2 def. = R k(x, y, x0, y0)d⇡(x0, y0) <latexit 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min ⇡1=↵,⇡2= R X⇥Y d(x, y)2d⇡ + " KL(⇡|↵ ⌦ 566 Extensions of Optimal Transport 566 Extensions of Optimal Transport Iterations <latexit 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⇡ <latexit 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DC-programming / Konno’s relaxation / Frank-Wolfe / . . .
  43. Open Problems! Toward high-dimensional OT: <latexit 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<latexit 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<latexit

    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<latexit 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<latexit 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! Geometrical properties of the Sinkhorn divergence ? <latexit sha1_base64="o6Pwb/wF7iyKddDfgh+wJSmcLxk=">AABB+HictVzNchu5EYY3f2vnz5scc5lEdsqbchRZ69rN1lYqK0uyrDVt0yYle3dpu4bkiBp7yKFnSMk2V2+SQ26pXPMCqco1eYG8QXLKK6R/gAGGxAwwimOUJAyIr7vRAzS6G6D70yTOZxsb/7zw3re+/Z3vfu/9i5e+/4Mf/ujHlz/4yWGezrNBdDBIkzR70g/zKIkn0cEsniXRk2kWheN+Ej3uv9zGzx+fRFkep5Pu7M00ejoOR5P4KB6EM2h6fvnjK70sHh3PwixLT68ErQg6h6N4MgrycZrOjidRngfxJDiGTsEwHkcTJPX755fXNtY36F+wWrkhK2tC/munHwQHoieGIhUDMRdjEYmJmEE9EaHIoXwtbogNMYW2p2IBbRnUYvo8EmfiEmDn0CuCHiG0voTfI3j6WrZO4Blp5oQeAJcEfjJABuIqYFLol0EduQX0+ZwoY2sV7QXRRNnewN++pDWG1pk4hlYXTvX0xeFYZuJI/JbGEMOYptSCoxtIKnPSCkoeGKOaAYUptGF9CJ9nUB8QUuk5IExOY0fdhvT5v6gntuLzQPadi3+TlFehBKIjR58WFEJxQvQDeptz+IzlSYDzCChEcoxYOyVdj2n0E+i/gPb7UM6opnTSh7Kg1rNa5DYUG3LbidyDYkPuOZEtKDZky4lsQ7Eh2xKJ2Ix0bsd3oNjwHSfnh1BsyIdO5CMoNuQjJ/IQig156ER+BcWG/MqJvA3FhrztRN6FYkPedSK7UGzIrhN5AMWGPHAid6HYkLsSWb1SMygp0Ykdq3IL6mUeaCkSaNlyyneLrKMNe8tjTQ8qsO5VvQN/7dgdD51GFdhdj3l3VIF1z7w9sJF2rNsW3aHdxIa948TuwwywY/ed2C/EiwrsFx4r7WUF1r3WWtDPjnVb33vwZMfec2LvQ82Ode9RD6DFjn3gsWNMK7BtJ/aheFWB9bH6WQXWbfc7YFfsWPc+1YX+dqyPNZ1XYN329BA8GDvWvVs9hlY79rET+0S8rsA+cWK/BOtux37pscO+rcCqPfYS7SAj8kciWLF11MJiVWJtCtRCB/+k2FsS8o370O7CjArMiDBjJ2KvQOx5IloFouUtV17Y0Zz8XTeXToHoeCL6xd6EtZmz/7Doj7XEA7FTIHaWEHUeKb5rNZYT8i5Uiws5K3YurPmMKS3sN9YiOR/qLa9CPCgheG4f08y/TtESRlCoqTpqx8Uez8iAnusQpxS9qVEqHm7crLAKJuq1E9W3oPpO1BsL6o0TNbeg5k7UiQV14kTplW/ieh4zQOsf38WCnngGsI9cXQLwCrZg17kDazSA+dMGL/ARtTyAvx2KvV2lTjKM5nGfxCzH05IlzqC2EGvQrqPCHYqvE1phEUjGPR/IGB+fMLexkGuOrfBZsZMHRcbEn05M8owKOugtBrSemtG5Sy1n5N1xrRn+TrHuVa0Zfpc0fkZePNea4WdS+tk5ZO9KbPcc2A6spqnUvq43pcH5F6ah6pdo10WLi291LOcM0nvdkP6+fDP753gv21Rj/eh6Mxq5Mb68NL4mNLSec0PPzaig98Rer6oFjUcykXGvrjeVIaVddCLl0E9N3wz2Gco3o+rNaLTB49qmmHth1JvO3mkxGl1vRuNQcN7zjDx5VW9GY0TPrA9db0YDsy2hjPN1vallRw1w7KzrTa36hLLAmAPiOc8t2ivKyE+aS2ox+Qf12RrT51/dxzBn86yIEeopad+2mk6/2MvqJVL+QgRWbdZQDvQv5oYPVqaxEJvO+IplmJX291U6eo9HzbdAiwGsfj4DcOXME5BQ5STQeidA8YYz6iqPTOE2nTicJUdLqJ5snTm9Rc2Xs0bltufU6orL9Gi1Hntkr3Oae1PyCVukWZceWpVvuIqiS0Otkobc9Jro7q1cr2Xtbzhx0yXEtJhpAzoR4pO0+jjVpvWOoeOr8pRnBoXPfPT8xWzzkbQ2GPOkZItQljqeZj+VRzLbcF+9LnSOmz8L6I2ivTohqxHTiVTujEJVtpi98QU9a9oHdCaHPJjGAN5jIKlMBZ+aYRYd8+kBWVTT3rp4o75Uho7rOVldZY/r0SMDPbKgm8c427Bj3IdaF2KGA3jqekQ5lwpdpaTxTPy6OB1N6Q3WR/RJyUIqGmxvopKFrIuyj0tUTgGNs4GjdH8ay3QUvrdCyR312+TRsWvZ8l+lk1t1vh3SHK+ezdWZmCFx3SSuAa0aPtXlp2UOLMHC+skm+a/1o0R+TTiiDXVxfWZwZr1M6MQ/ogh2Sp5xQqvNtTrKvc381PInilNbqLNzPM1OyUIGZP8C2J9SmpMB/Zh3B9QJOluEhGykj92JC+/G5uvEzjmm/bhY8K0GPd8ismVz4q/omqsrp7nIEQPvA2dLc1vppEW+YERcM2nd9dqu330Qqe9JmLOEKeq5co34f0i/1Y+aJ2srMwI1jG8gl7bO9j5SillQRyHt8vU2SPU1pbxSyPBMSq33Py3TlZJkOxRxoTy4Ww+B84CemRfOkozkzlf68D5al81FytMlPeJojyiKZ7s/kjswyn2ddsk1WnM9miUjmAWzIopQfV1Z5GW+9bzK1P1o5/8X6lrXZa0hxUDoDC5ryJXfjyhaM6VMYFbz/H1Jq8mu9WypVz2fCc3FsbGWv4HWn8NvJbd69qPTL1mFWzQHmIJ+0hrhlmClhx+vWyVeamYqWvpZ89NzUvUyW84TX7N10zH2SWMqbZo1r2XWQtXPQ+OFQeOFpw67dNaotajalSV67owtuvK00pdfE27dBpTnTspuj0yhYg8pzVjKj+rQSdUd4yvUWyetDSetEFareRpgrnkfpH2tL6/ub4rdPRC3ybcZkAfG8cuQVmlMPpdqrY/UmAJyvintq7n6e9SC3PtkQZEy3+PEFcOnTgMqZ4Wkv5Q7W0p2XlsEdW/pVPZRNrZH9Y9WkGNaEzmtS4W4ST0iKb8pR7BkkdYNnyOgzH9IPhX7HfUxs9lbv5Og5E/oeJNXlebFkcKE9O/KvO2vRK/7RvwaUEw4l951H2g1f8NIgTEqk2D3LHN6Q7jL8UkCe7R9sp+rdopP8SaGROsk9UL8zsPGcNSr57o5t9SI1dh+BT1R6/qt23q4+SXeHF38znOiF9KuNpY+6mLp+Xy0QrnLlZ/r9DBf4qv1Mac+ZmSho7wypic+8+bCEjXjwhgfLs1G0UT+ZpI3kZlPp3wpq96KcjnTwDbmmOIl1z1QRNi8u2tWb+5Dxzj6K/T6hDWpcYuLEmbjUpkfMC0tZqUuruxD3HqxdjdKjJ2oaqdQ1M3dQttvtpARWb9EuHI23NuUvVeKUtxZGKYwEHyjtyo+NGl+BgV/B8IWHSqOPrnDDvi3W2Jb7L6D2xCvZJ0zmgG1oC0YLsXeoRxnuUe9jl4Z1E36Phz8ecSga5f0Me2kTWVnym7JTer+9E/JCmQickqvezYfg8nFPZJVTk3GE5Nlc48mFuq7OE3Hojj4jKTMxZ8Pn2u4RnEk1Heamo1BUXePoMyhCQ91j8HvnevezXmZnOr1tcrFlwfvAurEReHw5K86VtH9fCxUZryRd88BrcNRDXW1W/yv41B8NKfmvHy55fRdsxceb537RTIji/5w8zWjufnM5mqO/jzTYnTaW7LzY78vaPSmUmM0754++qN6DiheC8F5ULd0jDdnkZbXlwqeC9hkSMV/xF8vuL+N8KqgUSVHE0rqnKKamurhpqa+cWkbnfrMRyZNp0qmMjUdR3ToRuy22Be34We78ACb3g7l71LyX8Tavz87hNYjsh4qi86Zgx61RZT90KdoQ3rW92erJMa7vHy3twsteBbeola853uf+uNd325pbNXfIOG1fk+kYliKSJZP9/S66sMIyidvnANS3/MN6C49Z7H45tnY42xR3Z9almhBn7hvFvQr8X1DygHN1ak8q8eTA7xhHxb5oUD8htpCaedxz3Vxbldybi9xzkk7ZQ6vjc/q72ZVcdk2uAyL3NmJ7JdSnK3P8+pzozuVXPgOej1+VIMfGVJ2SPsvKRLORH02b15Dcy5lMk9YJ0JlIlkPGGeGxfuuj2xPanideIz/biX6riHpHsjSp/x3QCdsGdFLpG52SXq+6VifSb1TI638HiX97waf0r+AK5/clJVPbxT/u8Hh5vqNj9c/eri59vkt+f8cvC9+Jn4hrsEa/0R8DtTa4gA4/EH8Tfxd/GPr7dYft/609Wfu+t4FifmpKP3b+st/Abq/t10=</latexit> ! Leveraging smoothness in high dimension?
  44. Open Problems! Toward high-dimensional OT: <latexit 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<latexit 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<latexit

    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<latexit 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<latexit 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Gromov Wasserstein: <latexit 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! Existence of Monge maps for " = 0? <latexit sha1_base64="0C/SfsK2UN+57vTCu3DbXeiRgIo=">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</latexit> ! Taylor expansions when " ! 0 X <latexit 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! Geometrical properties of the Sinkhorn divergence ? <latexit sha1_base64="o6Pwb/wF7iyKddDfgh+wJSmcLxk=">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</latexit> ! Leveraging smoothness in high dimension?