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A Schrödinger Tour of Data Sciences

8f7426d5d9183245fe9b02d1e972a597?s=47 Gabriel Peyré
September 16, 2021
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A Schrödinger Tour of Data Sciences

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

8f7426d5d9183245fe9b02d1e972a597?s=128

Gabriel Peyré

September 16, 2021
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Transcript

  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=">AAA/fHiczVvddhu3EUbSv1j9s9vLXnRbxY2dOj6y6tO0J6d1bEmWFdOWbIm2kzB2+bOiNl5x6V1Ssswyb9Lb9qKP0CfpG9SP0Kt2ZgAssCR2B3DTcyIcSVgQ38xgAMwPsOyN06SYrK398623v/Xt73z3e++cW/n+D374ox+fv/CTR0U2zftxu5+lWf6k1y3iNBnF7UkySeMn4zzuHvfS+HHv+QZ+/vgkzoskGx1MzsbxF8fd4Sg5TPrdCTQ9O39h1iEis146jeedXjx/dn517eoa/UTLlWuqsirUz1524efvi44YiEz0xVQci1iMxATqqeiKAsrn4ppYE2No+0LMoC2HWkKfx2IuVgA7hV4x9OhC63P4O4Snz1XrCJ6RZkHoPnBJ4TcHZCQuAiaDfjnUkVtEn0+JMrbW0Z4RTZTtDP73FK1jaJ2II2jlcLqnP24ArYcgG2qmecwT6Pc7GmsCvcfUglroK25T0h6OMLJGPwEKY2jD+gA+z6HeJ6Sej4gwBekI56BLn/+LemIrPvdV36l4TaP5pujvm6eXiwElEjvivtgQLdEWm2JL7JO8zWWFcBswjjHoOAfJh6AhlPYSSHMZ/q/DnloTH0JtG2TsUZ8YxhKJPfh/RvKvKFkjcU/cFAfiDvG+BPV7ULscME9cT722C5oN/3WzwqyynGbkrFGCCY1/ShYgFy8b++LaGajZPKT1UKgd2TynUoMb4qHYbZw/5DwC+qdkiY5Jg8hxBu1DkAlXaAr80CbiWv67+EqsQv0r8Q+1jiOQK1cr+4joySd/mn+CgjTfgzIX54hmG6ReKUcZwZOcgaxc/V1xQnsjIks+hc/kXkpp7Y2oV9Po7kOZU02vmh6UGbXOG5EbUFzIDRa5DcWF3GaRLSguZItF7kFxIfcUsgn7AIoL+4Dl+hCKC/mQRR5AcSEPWOQjKC7kIxb5GRQX8jMWeRuKC3mbRd6F4kLeZZFtKC5km0VuQXEhtxSyfq+hXcuITsLsq5tQr/JAP5VCy01WvltkY13YWx67sl+D5fflJvx3Yzc9dBrXYLc8Vs9hDZZfP9tg5dxY3prcIR/pwt5hsTuwAtzYHRb7ifiyBvuJx355XoPld0yL/I0Ly9vPe/Dkxt5jsfeh5sbyXmYXWtzYXQ+bP67B7rHYB+JFDdbH6uc1WN7u71Ms5sLue/iMSQ2W9xptiB3cWN6ePoIYxI3lfc5jaHVjH7PYJxQ1urBPWOynFJ26sJ96+MlXNVjtKWX0PqQYMIYd20StW+5KrI2BWpfhn5a+BWvoowYsZlhihoQ5ZhHbJWLbE9EqES1vuYrSjhYUsfJc9kvEvieiV/omrE3Y/oOy/4ByMx6xWSI2FxBNWQDOtR7LCUUXuoVDTkrPhTWfMWWl/cZarNZDs+XViN0KQq7tI1r5VyhXx0xmQHlrPbWj0sdLZETPTYhTyvX0KDUPHjcprYKNesmieg5Uj0WdOVBnLGrqQE1Z1IkDdcKizM63cR2PFWD0j3Mxoyed+3NnKdWzi13wuFvg/bBlF/77nKU0Z+V4IoCeUmfPxhbnUJtRJm0yu03KkeVJQwySyZ676owJn/BscqZ2nbTD89KXR0KfePrTSUieYUkH48WIdlQYnbvUMqf4TtbC8HfKna9rYfgt0vic4nhZC8NPlPSTN5D9QGEP3gC7D/tprLRv6qE05BmKpKHrnG3eKe0mnle+pJ0j20L5b1BN6sDUw2gU1hiKyhhCaBhdFpYuw6hgjCRjW12LgkcyUtmtqYfKkJGvHCk5zFPozGCfgZoZXQ+jsQdx1QZl1jOrHrpCx+VoTD2MxiMhz9bnFK/rehiNIT1LfZh6GA08U+mqbN7UQ603akBmyKauY5acohh94pyQ924+S7Ej8mUfgycqT8sIvpmSiTzr6fRKP9Ms0aJ1CZEDvf/UipCqNGZinc1+pAyTiu9dpmP8L2q+BVqMYNfKc33uTDoFCfWJQUyn5E+JWjOmOjKNW2dxaEkOF1Ad1TphYznDV57pVNueUSuXNZnRGj12yM4WtPbGFLG1SLOcHlq1M1xHkdNQq6Ihnl6I7l7RDs4WtL/G4sYLiHG50vp0OyRvqZuzSJfW9y0dX1S3KBMo8k7FrF88Cz4kXE4ZSUbWBmVp4mn306c8dhv6wyvCnEDLzyKaUbRXJ2Q1ErrxKdg4RJ/lykh5Rs+Gdpvu2aq3VpGiMhbyRhXPuPG0O6L7TNvWcrxRX/r8TNYLsrrmHr35dsyghw50aPaxAb7iPtQOIJpvw9OBR/5h7twy0ncuPijvPzOav+Zs277Z65Q0pLWJK/axKQM+qlA5BTSuBZlB+9NYpKPxnSVKfEbuksfklVW7f5Hu9PWbI11a4fVruf6UZEBc14lrRHtG3vfLp0UOUoKZ85N1ijqbR4n8QjiiBeW4PrU4S72M6F2amHLLMcWzKe01bm9Ue9tnR4ufaE57Qr9VgXfFGdnHiKxfBN4pozUZ0a/9Vo5+t0Lag5QspI/VScrYxhXpJOwaS2iXyzUi3xcy6y0mSzYl/pquvbsKWosyzpdeYL6wtrVOWhQJxsQ1V7bd7O1m34NI8yaEvUokRbNWLhH/y/RX/+p1srq0IlDDOAOFsnSu+cgo00AddcnHN9sg3deW8t1ShqdKauP9jEzvViTbpDwJ5UFfPQDOfXqWvHCV5CR3sdRHetGmk1akPF7QI44WvWtHWf2h8r8o9xXykau05zrCvGGjY3/dlzvhXeTbzKtK3Y928X+hbnRd1RpSjIQ5XZUa4s7eY8qxbClTeidJvn0TK0rLWs8XejXzGdFaPLb28p+h9RfwV8utn/3o9CpW4RatAUnBPBmNyJZoqYcfr1sVXnplalrm2fAza1L3slveJCuW1s1kxifBVPZo1bxUZw267jP+A7rDMxrQ7dqKPFuIoA/UnZ8v9TDa/pSnLGU+dtKoxENKO+fxozpgqfK5uEa9YmmtsbTwbTz7RN3enbbvvE2xQ58iHJkdDGgXJBTT6NbmPEhSQF7Xlf2yd1eHWnAv9chCIeXqW4V4dtcX8h1SKeOvlOfIyI6aHaff2TlVfbQN61D9N0vIY8pmC9o7GnGdesRKfluOaGHHX7V8ekRn3l2KWaRfb85I7d5mFqKKvzbZnNwLhlcq3lPR+IjmYCJeN3LbWcoPd6wMMRLyvi8tI3ic5ddBs4x5m1wZOld3R28FzRJ6kpzyeRk19shGLZ8yyFusEWhfr7urJPVM/MHDOsi80uwUe33hZ1OVleDY3oeeqHkz864ePL/UmyPH703us7rkOY5VHDhbeH4zWl3lSarPTXqYLvA1+piKkcoJdfRuMqkqpiM+8uYiJQrjIjE+XMJGESJ/mOQhMst7G1/KuremXM3mpY05opyEew8SEa4I6pIzYrrMjKO3RK9HWJuabOEo4XlXpnJw29riyc+5JV8kW881eqTU8kZ13kJTtz2GseHSQsZk/VLBnYvI3n0h30ety6Ds3OAjKPg3Eq78SXsTn9O1fYgi8b16v29FNN/kv1B1eeIXUQvu5MFCdtpV46z2aJ7lFxZ1m74PB38eCeiakz4hPxgqu6TMS25T96d/Sns4FzErvekZPgabCz+SZU4h40nILvGjSYT+HlPoWDQHn5FUufjzkef+3CgOhf4+XdgYNHV+BFUOITz0/bzfnJve4bxsTs36Wubiy0PacH0joXF4M1afbZh+PhYqt2bk6+eA1uGwgbr2Fv/rODQfwymcly+3gr7r9KXHrMt+sTqzxGg2fM8Ybj6ruZ6jP8+sHJ2Jddz8ZNQWBc1UZo3m66eP0aRZA5rXTMiTQl46ibdXkZHXlwqenLtkyMS/vWSQ+DoZfKnoE/x6SroHT01/0881Kv2Zj0yGTp1MVWpIT2baHdrDMZ0WmJudAT2Xb1s+O796bfFb58uV9vrV31+99uD66se31BfS3xE/E7+EQP6a+FB8LO5AgNkGEU7FX8Rfxd/++J8bF2/8+sYHsuvbbynMT0Xl58Zv/wsjoSbD</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=">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</latexit> <latexit 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<latexit 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<latexit 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↵✓ <latexit 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min ✓ Wp ",p (↵✓, ) <latexit 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  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=">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</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=">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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.:
  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=">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</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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</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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