Numerical
Optimal Transport
http://optimaltransport.github.io

Density Fitting
Gabriel Peyré
www.numerical-tours.com
ÉCOLE NORMALES U P É R I E U R E

Weak vs Strong Topology
Random vectors

Radon measures

P(X 2 A)

R

=

A

d↵(x)

Weak⇤ convergence:

Convergence in law:

8 set A
n!+1
P(Xn 2 A) ! P(X 2 A)

8 continuous function f

R

f d↵n

n!+1 R!

Weak convergence:
...

↵1

↵2

↵3

...

↵

Key question: quantifying weak convergence.
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Central limit theorem: If E(X) = 0, E(X 2 ) = 1 and (Xi )i i.i.d.
⇠ X
def. X1 + . . . + Xn lawpYn =! N (0, 1)nCentral Limit Theorem
Central limit theorem: If E(X) = 0, E(X 2 ) = 1 and (Xi )i i.i.d.
⇠ X
def. X1 + . . . + Xn lawpYn =! N (0, 1)nKolmogorov-Smirnov distance: dKS (X, Y ) = max |P(X 6 t)def.

tP(Y 6 t)|law

Metrizes convergence in law: X ! Y , dKS (X, Y ) ! 0
Theorem:Carl-Gustav Esseen[Berry 1941][Esseen, 1942]

CE(|X|3 )pdKS (Yn , N (0, 1)) 6nC 6 1/2

n = 1 P(Yn 6 t)
n=4
n=7

t

Central Limit Theorem
Central limit theorem: If E(X) = 0, E(X 2 ) = 1 and (Xi )i i.i.d.
⇠ X
def. X1 + . . . + Xn lawpYn =! N (0, 1)nKolmogorov-Smirnov distance: dKS (X, Y ) = max |P(X 6 t)def.

tP(Y 6 t)|law

Metrizes convergence in law: X ! Y , dKS (X, Y ) ! 0
Theorem:Carl-Gustav Esseen[Berry 1941][Esseen, 1942]

CE(|X|3 )pdKS (Yn , N (0, 1)) 6nC 6 1/2

n = 1 P(Yn 6 t)
n=4
n=7

t

Multi-dimensional extension: use W1 in place of dKS !
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Overview• Csiszar Divergences• Dual Norms and MMD• Minimum Kantorovitch Estimators• Deep Generative Models Fitting

Strong Norms
Reference measure dx on X .
<latexit sha1_base64="a44VEdhkyzoc7MA5hkasuMslSOQ=">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</latexit><latexit
sha1_base64="a44VEdhkyzoc7MA5hkasuMslSOQ=">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</latexit>

Lp norms on
densities:✓<latexit sha1_base64="2JD/jKR1N5jqzrWaJC57kEeEWIc=">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D(↵, ) =<latexit sha1_base64="MRXohxEFGoI25S4PMP017weYREo=">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</latexit><latexit
sha1_base64="MRXohxEFGoI25S4PMP017weYREo=">AABCXnictVxfcxu3EYeStonVpnHal870Ba3ijpRxHEnxTNLJZCbWH8uKaZs2KdtJaGnuyCNN+8ij70hZNsOP0s/Sz9HHvrVP/QrdXQAHHIk74FTXGEk4EL/dxR6w2F2ADifxMJtub/9j7b33f/HLX33w4ZX1X//mo99+fPWT3z3OklnajU66SZykT8Mgi+LhODqZDqdx9HSSRsEojKMn4ct9/PzJeZRmw2Tcnr6ZRM9GwWA87A+7wRSazq7+7WCzE8TXO2G0xTvRq17U551JMOed4Xh61nl6wTd5p58G3XmnB/0W8IdfLDYvtj7PW8NIt/Kt0wnHB744ne98MVnwb/l6Z5yko3C+vkrHQmR9cTZvnE428WFrcXZ1Y/vGNv3jq5UdWdlg8l8z+YSfsA7rsYR12YyNWMTGbAr1mAUsg/IT22HbbAJtz9gc2lKoDenziC3YOmBn0CuCHgG0voTfA3j6SbaO4RlpZoTuApcYflJAcnYNMAn0S6GO3Dh9PiPK2FpGe040UbY38DeUtEbQOmXPodWFUz19cTiWKeuzr2kMQxjThFpwdF1JZUZaQcm5MaopUJhAG9Z78HkK9S4hlZ45YTIaO+o2oM//RT2xFZ+7su+M/ZukvAaFs5YcfZJTCNg50ef0NmfwmZAnBs4DoBDJMWLtNel6RKMfQ/85tN+HsqCa0kkIZU6ti0rkPhQbct+JPIJiQx45kQ0oNmTDiWxCsSGbEonYlHRux7eg2PAtJ+eHUGzIh07kIyg25CMn8jEUG/KxE/kjFBvyRyfyNhQb8rYTeReKDXnXiWxDsSHbTuQJFBvyxIk8hGJDHkpk+UpNoSREZ+hYlbegXuSBliKGlltO+fbIOtqwex5ruluCda/qA/hrxx546DQqwR56zLt+CdY9847ARtqxblt0h3YTG/aOE3sMM8COPXZiv2cvSrDfe6y0lyVY91prQD871m1978GTHXvPib0PNTvWvUc9gBY79oHHjjEpwTad2IfsVQnWx+qnJVi33W+BXbFj3ftUG/rbsT7WdFaCddvTx+DB2LHu3eoJtNqxT5zYp+yiBPvUif0BrLsd+4PHDvu2BKv22HXaQQbkj0SwYquoBfmqxNoEqAUO/nG+t8TkG4fQ7sIMcsyAMCMn4ihHHHkiGjmi4S1XltvRjPxdN5dWjmh5IsJ8b8La1Nm/l/fHWuyBOMgRB0uIKo8U37Uayzl5F6rFhZzmOxfWfMaU5PYba5GcD9WWVyEeFBBibj+nmX+doiWMoFBTVdSe53u8QHJ6rkK8puhNjVLxcOOmuVUwURdOVGhBhU7UGwvqjRM1s6BmTtS5BXXuROmVb+I6HjNA6x/fxZyexAwQPnJ54eAV3IJd5w6sUQ7zpwle4CNqeQB/WxR7u0qVZBjN4z6JWY5nBUucQm3ONqBdR4UHFF/HtMIikEz0fCBjfHzC3MZcrjlhhRf5Ts7zjIk/nSHJM8jpoLfIaT3Vo3OXWhbk3YlaPfydfN2rWj38IWl8QV68qNXDT6X000vI3pbY9iWwLVhNE6l9Xa9LQ+RfBA1VX6ddFy0uvtWRnDNI76Im/WP5Zo4v8V72qSb0o+v1aGTG+LLC+OrQ0HrODD3Xo4Lek/B6VY3XHslYxr26XleGhHbRsZRDP9V9M9inJ9+Mqtej0QSPa59i7rlRrzt7J/lodL0ejcdM5D0X5Mmrej0aA3oW+tD1ejQw2xLIOF/X61p21ICInXW9rlUfUxYYc0BizosW7RWl5CfNJLUh+QfV2RrT51/dxzBnc5rHCNWUtG9bTifM97JqiZS/EIFVm9aUA/2LmeGDFWnM2a4zvhIyTAv7+yodvcej5hugRQ6rX5wBuHLmMUiochJovWOguOOMuoojU7hdJw5nSX8J1ZGtU6e3qPmKrFGx7YxaXXGZHq3WY4fsdUZzb0I+YYM069JDo/QNl1F0aahR0JCbXh3dvZXrtaj9bSdusoSY5DOtSydC4iStOk61ab1l6PiaPOWZQhFnPnr+Yra5L60NxjwJ2SKUpYqn2U/lkcw23FevM53jFp9xeqNor87JagzpRCpzRqEqWyy88Tk9a9ondCaHPASNLrxHLqlMmDg1wyw65tM5WVTT3rp4o75Uhk7UM7K6yh5XowcGemBB149x9mHHuA+1NsQMJ/DU9ohy1nNdJaTxlH2en44m9AarI/q4YCEVDWFvooKFrIqynxeovAY0zgYRpfvTWKaj8J0VSu6o3yaPjl2Llv8andyq8+2A5nj5bC7PxPSI6y5x5bRqxKmueFrmICSYWz/ZJf+1epTIrw5HtKEurqcGZ6GXMZ34RxTBTsgzjmm1uVZHsbeZn1r+RHFqMnV2jqfZCVlITvaPw/6U0Jzk9GPeHVAn6MIixGQjfezOMPdubL7O0DnHtB83ZOJWg55vEdmyGfFXdM3VldFcFBGD2AcWS3Nb6aRBvmBEXFNp3fXart59EKnvSZizRFDUc2WT+G/Rb/Wj5snGyoxADeMbyKSts72PhGIW1FFAu3y1DVJ9TSk/zWU4lVLr/U/L9GlBsgOKuFAe3K17wLlLz4IXzpKU5M5W+oh9tCqbi5QnS3rE0fYpihd2fyB3YJT7Ou2SG7TmOjRLBjALpnkUofq6ssjLfKt5Fan70c7+L9S1rotaQ4qc6Qyu0JArvx9RtGZKGcOsFvP3Ja0mu9bTpV7VfMY0F0fGWv4ZWv8Ev5Xc6tmPTliwCns0BwQF/aQ1Ilr4Sg8/XnsFXmpmKlr6WfPTc1L1MlsuE18L66Zj7PPaVJo0ay5k1kLVL0PjhUHjhacO23TWqLWo2pUlOnPGFm15WunLrw63dg3KMydlt0emUEMPKc1Yyo9qz0nVHeMr1FsnrW0nrQBWq3kaYK55H6R9rS+v7p/z3Z2z2+TbdMkDE/FLj1bpkHwu1VodqQkKyPmmtK/m6u9QC3IPyYIiZXGPE1eMOHXqUlnkkv5F7mwJ2XltEdS9pdeyj7KxHap/uYIc0ZrIaF0qxE3qEUn5TTn4kkW6YfgcnDL/AflUwu+ojpnN3vqd8II/oeNNsao0LxEpjEn/rszb8Ur0emzEr5xiwpn0rkOgVf8NIwWBUZkEu2eZ0RvCXU6cJAiPNiT7uWqnxCne2JDoBkk9Z9962BgR9eq5bs4tNWI1ts+gJ2pdv3VbDze/2Juji99lTvQC2tVG0kedLz1fjlYgd7nic5UeZkt8tT5m1MeMLHSUV8R02DfeXIRE9bgIjA+XeqOoI389yevILE6nfCmr3opyMdMgbMxzipdc90ARYfPuNq3e3JZjHOEKvZCwJjXR4qKE2bhE5gdMS4tZqSsr+5BovVK5G8XGTlS2Uyjq5m6h7bewkBFZv5i5cjaityl7pxCluLMwgkKXiRu9ZfGhSfMbKPibM1t0qDj65A5b4N/eYvvs8B3chngl6yKjyakFbUFvKfYO5DiLPap19MqgbtL34eDPYwi6dkk/pJ20ruyCsltyk7o//ddkBVIWOaXXPeuPweTiHskqpzrjGZJlc49myNR3ceqORXHwGUmRiz8fca7hGkWfqe801RuDou4eQZFDHR7qHoPfO9e96/MyOVXra5WLLw+xC6gTF4XDk7/yWEX387FQqfFG3j0HtA79Cupqt/hfx6H4aE71eflyy+i7Zi883rroF8mMLPrD9deM5uYzm8s5+vNM8tFpb8nOT/h9vNabSozRvHv66I/qOaB4zZnIg7qlE3hzFml5fanguYBNhoT9h/19zf1thFc5jTI56lBS5xTl1FQPNzX1jUvb6NRnPjJpOmUyFanpOKJFN2L32TG7DT/7uQdY93ao+C6l+ItY+/dne9DaJ+uhsugic9ChtoiyH/oUrUfP+v5smcR4l1fc7W1DC56FN6gV7/nep/5417ddGFv5N0jEWr/HEtYrRCTLp3t6XYUwguLJm8gBqe/5crpLL7JY4ubZyONsUd2fWpZoTp+4bxaEpfjQkLJLc3Uiz+rx5ABv2Ad5foizL6gtkHYe91wX52Yp5+YS54y0U+RwYXxWfTerjMu+waWX587OZb+E4mx9nledGz0o5SLuoFfjBxX4gSFli7T/kiLhlFVn82YVNGdSJvOEdcxUJlLoAePMIH/f1ZHteQWvc4/x3y1F3zUkPQJZQsp/czphS4leLHVzSNKLm47VmdQ7FdLK71GeXd3YWf6/DFYrJ7s3/npj5+HNje/25H9z8CH7I/sz24Ql/hX7Dog12Qnrrr2/trW2u/blrX/ufbD30d7Hout7axLze1b4t/eH/wLtV9VW</latexit>

Z

d↵
( (x)
X dx

d
(x))p dx
dx

! defined only if ↵ ⌧ dx and

◆1/p

d↵
=
dx

⌧ dx.

d
dx Lp (dx)Strong Norms
Reference measure dx on X .
<latexit sha1_base64="a44VEdhkyzoc7MA5hkasuMslSOQ=">AABB5HictVzNchTJES7Wfwv+Y+2jL20LHKwDy0LeCNux4YgVkhBaBAhmBOzuADE/rWGgZ3ronhGCWT2ADz45fPUr+Gqf/Bx+A/vkV3D+VHVVz1R3VsuYCknVNfVlZmVXZWVm1dCbJqN8trHxzwsffOOb3/r2dz68eOm73/v+D354+aMfPcrTedaPj/ppkmZPet08TkaT+Gg2miXxk2kWd8e9JH7ce7WNnz8+ibN8lE7as7fT+Om4O5yMjkf97gyanl9eexgfx1k86cfROO7m8yyOrnQG0emVKJ1A7cnplfUIem2sb9C/aLVyQ1fWlP53mH4UHamOGqhU9dVcjVWsJmoG9UR1VQ7lK3VDbagptD1VC2jLoDaiz2N1pi4Bdg69YujRhdZX8HsIT1/p1gk8I82c0H3gksBPBshIXQVMCv0yqCO3iD6fE2VsraK9IJoo21v429O0xtA6Uy+gVcKZnqE4HMtMHavf0hhGMKYpteDo+prKnLSCkkfOqGZAYQptWB/A5xnU+4Q0eo4Ik9PYUbdd+vxf1BNb8bmv+87Vv0nKq1Ai1dKjTwsKXXVC9CN6m3P4jOVJgPMQKMR6jFh7Q7oe0+gn0H8B7fegnFHN6KQHZUGtZ7XIbSg+5LaI3IPiQ+6JyAMoPuSBiDyE4kMeaiRiM9K5H9+C4sO3RM4PoPiQD0TkQyg+5EMR+QiKD/lIRH4JxYf8UkTeguJD3hKRd6D4kHdEZBuKD9kWkUdQfMgjEbkLxYfc1cjqlZpBSYnOSFiVW1Av80BLkUDLlijfTbKOPuzNgDXdr8DKq3oH/vqxOwE6jSuwuwHz7rgCK8+8PbCRfqxsi27TbuLD3hax+zAD/Nh9Efu5elmB/Txgpb2qwMpr7QD6+bGy9b0LT37sXRF7D2p+rLxH3YcWP/Z+wI4xrcAeitgH6nUFNsTqZxVY2e63wK74sfI+1Yb+fmyINZ1XYGV7+gg8GD9W3q0eQ6sf+1jEPlGnFdgnIvYLsO5+7BcBO+y7CqzZYy/RDjIkfySGFVtHrVusSqxNgVpX4J8Ue0tCvnEP2iXMsMAMCTMWEXsFYi8QcVAgDoLlygs7mpO/K3NpFYhWIKJX7E1Ym4n9B0V/rCUBiJ0CsbOEqPNI8V2bsZyQd2FaJOSs2LmwFjKmtLDfWIv1fKi3vAZxv4Tguf2CZv51ipYwgkJN1VF7UezxjIzouQ7xhqI3M0rDQ8bNCqvgok5FVM+D6omotx7UWxE196DmIurEgzoRUXblu7hOwAyw+sd3saAnngHsI1eXCLyCLdh1bsMajWD+HIIX+JBa7sPfFsXeUqmTDKN53Ccxy/G0ZIkzqC3UGrTbqHCH4uuEVlgMknHP+zrGxyfMbSz0mmMrfFbs5FGRMQmnMyJ5hgUd9BYjWk/N6NyhljPy7rjWDH+7WPem1gy/Sxo/Iy+ea83wMy397ByytzW2fQ5sC1bTVGvf1pvS4PwL0zD1S7TrosXFtzrWcwbpnTakv6/fzP453ss21Vg/tt6MRu6MLy+NrwkNq+fc0XMzKug9sddralHjkUx03GvrTWVIaRedaDnsU9M3g30G+s2YejMah+BxbVPMvXDqTWfvtBiNrTej8Uhx3vOMPHlTb0ZjSM+sD1tvRgOzLV0d59t6U8uOGuDY2dabWvUJZYExB8RznlusV5SRnzTX1EbkH9Rna1yff3Ufw5zNsyJGqKdkfdtqOr1iL6uXyPgLMVi1WUM50L+YOz5YmcZCbYrxFcswK+3vq3TsHo+aPwAtRrD6+QxAypknIKHJSaD1ToDiDTHqKo/M4DZFHM6S4yVUR7fORG/R8uWsUbntObVKcZkdrdVjh+x1TnNvSj7hAWlW0sNB5Ruuoihp6KCkIZleE9290+u1rP0NETddQkyLmdanEyE+SauPU31abzk6vqpPeWZQ+MzHzl/MNh9ra4MxT0q2CGWp4+n2M3kktw331evK5rj5s4jeKNqrE7IaIzqRysUo1GSL2Rtf0LOlfURncsiDafThPUaaylTxqRlm0TGfHpFFde2txBv1ZTJ0XM/J6hp7XI8eOuihB908xtmGHeMe1NoQMxzBUzsgyrlU6ColjWfql8XpaEpvsD6iT0oW0tBgexOXLGRdlP2iROUNoHE2cJQeTmOZjsF3VijJUb9PHhu7li3/VTq5NefbXZrj1bO5OhMzIK6bxDWiVcOnuvy0zIElWHg/2ST/tX6UyK8JR7ShEtdnDmfWy4RO/GOKYKfkGSe02qTVUe7t5qeWPzGcDpU5O8fT7JQsZET2L4L9KaU5GdGPe3fAnKCzRUjIRobYnVHh3fh8nZE4x6wfN1J8q8HOt5hs2Zz4G7ru6sppLnLEwPvA2dLcNjo5IF8wJq6Ztu52bdfvPoi09yTcWcIU7Vy5Rvw/pt/mx8yTtZUZgRrGN5BrW+d7HynFLKijLu3y9TbI9HWlvFLI8ExLbfc/K9OVkmQ7FHGhPLhbD4Bzn56ZF86SjOTOV/rwPlqXzUXK0yU94miPKYpnuz/UOzDKfZ12yTVacx2aJUOYBbMiijB9pSzyMt96XmXqYbTz/wt1q+uy1pBipGwGlzUk5fdjitZcKROY1Tx/X9Fq8ms9W+pVz2dCc3HsrOWvofWn8NvIbZ7D6PRKVuEmzQGmYJ+sRrglWukRxutmiZeZmYaWfbb87Jw0vdyW88TXbN1sjH3SmMohzZpTnbUw9fPQeOnQeBmowzadNVotmnZjiZ6LsUVbn1aG8mvCrd2A8lykLHtkBjUKkNKNpcKoDkSqcoxvUO9EWhsirS6sVvc0wF3zIUj/Wl9e3V8Xu3ukbpFv0ycPjOOXAa3SEflcprU+UmMKyPkTbV/d1d+hFuTeIwuKlPkeJ64YPnXqUzkrJP253tlSsvPWIph7S290H2NjO1T/9QpyTGsip3VpEJ9Qj1jL78oRLVmkdcfniCjz3yWfiv2O+pjZ7W3fSVTyJ2y8yavK8uJIYUL6lzJv+yvR674Tv0YUE861d90DWs3fMFJgjMkk+D3LnN4Q7nJ8ksAebY/s56qd4lO8iSPROkm9UL8PsDEc9dq57s4tM2Iztl9AT9S6feu+HjK/JJijxO88J3pd2tXG2kddLD2fj1ZX73Ll5zo9zJf4Wn3MqY8bWdgor4zpqE+DubBEzbgwJoRLs1E0kb+Z5E1k5tOpUMqmt6FczjSwjXlB8ZJ0DxQRPu/umteb+1gYR2+FXo+wLjVukShhNi7V+QHX0mJW6uLKPsStF2t3o8TZiap2CkPd3S2s/WYLGZP1S5SUs+HeruydUpQiZ2GYQl/xjd6q+NCl+SkU/B0pX3RoOIbkDlvg326pbbX7Hm5DvNZ1zmhG1IK2YLAUe3f1OMs96nX02qHu0g/hEM5jBLqWpB/RTtpUdqYsS+5SD6f/hqxApmJRetuz+RhcLvJIVjk1Gc+ILJs8mpEy38VpOhbDIWQkZS7hfPhcQxrFsTLfaWo2BkNdHkGZQxMe5h5D2Du3vZvzcjnV62uVSygP3gXMiYvB4clfdaxi+4VYqMx5I++fA1qH4xrqZrf4X8dh+FhOzXmFcsvpu2YvA94694t1Rhb94eZrxnILmc3VHMN5psXorLfk58d+X9ToTaXOaN4/ffRH7RwwvBaK86CydIx3Z5GVN5QKngv4ZEjVf9Q/LsjfRnhd0KiSowklc05RTc30kKmZb1z6Rmc+C5HJ0qmSqUzNxhEtuhG7rfbVLfjZLjzAprdD+buU/Bex/u/PDqD1mKyHyaJz5qBDbTFlP+wp2oCe7f3ZKonxLi/f7W1DC56FH1Ar3vO9R/3xrm+7NLbqb5DwWr+rUjUoRSTLp3t2XfVgBOWTN84Bme/5RnSXnrNYfPNsHHC2aO5PLUu0oE/kmwW9SnzPkbJPc3Wqz+rx5ABv2HeL/FCkfkVtXW3ncc+VOB9Wcj5c4pyTdsocTp3P6u9mVXHZdrgMitzZie6XUpxtz/Pqc6M7lVz4Dno9fliDHzpStkj7rygSzlR9Nm9eQ3OuZXJPWCfKZCJZDxhndov3XR/ZntTwOgkY/51K9B1H0j2QpUf574hO2DKil2jd7JL0fNOxPpN6u0Za/T3K55fXbiz/XwarlaPN9d+t33iwufbZTf3fHHyofqJ+pq7BEv+N+gyIHaojYPAH9Vf1N/X3reHWH7f+tPVn7vrBBY35sSr92/rLfwFs2ay7</latexit><latexit
sha1_base64="a44VEdhkyzoc7MA5hkasuMslSOQ=">AABB5HictVzNchTJES7Wfwv+Y+2jL20LHKwDy0LeCNux4YgVkhBaBAhmBOzuADE/rWGgZ3ronhGCWT2ADz45fPUr+Gqf/Bx+A/vkV3D+VHVVz1R3VsuYCknVNfVlZmVXZWVm1dCbJqN8trHxzwsffOOb3/r2dz68eOm73/v+D354+aMfPcrTedaPj/ppkmZPet08TkaT+Gg2miXxk2kWd8e9JH7ce7WNnz8+ibN8lE7as7fT+Om4O5yMjkf97gyanl9eexgfx1k86cfROO7m8yyOrnQG0emVKJ1A7cnplfUIem2sb9C/aLVyQ1fWlP53mH4UHamOGqhU9dVcjVWsJmoG9UR1VQ7lK3VDbagptD1VC2jLoDaiz2N1pi4Bdg69YujRhdZX8HsIT1/p1gk8I82c0H3gksBPBshIXQVMCv0yqCO3iD6fE2VsraK9IJoo21v429O0xtA6Uy+gVcKZnqE4HMtMHavf0hhGMKYpteDo+prKnLSCkkfOqGZAYQptWB/A5xnU+4Q0eo4Ik9PYUbdd+vxf1BNb8bmv+87Vv0nKq1Ai1dKjTwsKXXVC9CN6m3P4jOVJgPMQKMR6jFh7Q7oe0+gn0H8B7fegnFHN6KQHZUGtZ7XIbSg+5LaI3IPiQ+6JyAMoPuSBiDyE4kMeaiRiM9K5H9+C4sO3RM4PoPiQD0TkQyg+5EMR+QiKD/lIRH4JxYf8UkTeguJD3hKRd6D4kHdEZBuKD9kWkUdQfMgjEbkLxYfc1cjqlZpBSYnOSFiVW1Av80BLkUDLlijfTbKOPuzNgDXdr8DKq3oH/vqxOwE6jSuwuwHz7rgCK8+8PbCRfqxsi27TbuLD3hax+zAD/Nh9Efu5elmB/Txgpb2qwMpr7QD6+bGy9b0LT37sXRF7D2p+rLxH3YcWP/Z+wI4xrcAeitgH6nUFNsTqZxVY2e63wK74sfI+1Yb+fmyINZ1XYGV7+gg8GD9W3q0eQ6sf+1jEPlGnFdgnIvYLsO5+7BcBO+y7CqzZYy/RDjIkfySGFVtHrVusSqxNgVpX4J8Ue0tCvnEP2iXMsMAMCTMWEXsFYi8QcVAgDoLlygs7mpO/K3NpFYhWIKJX7E1Ym4n9B0V/rCUBiJ0CsbOEqPNI8V2bsZyQd2FaJOSs2LmwFjKmtLDfWIv1fKi3vAZxv4Tguf2CZv51ipYwgkJN1VF7UezxjIzouQ7xhqI3M0rDQ8bNCqvgok5FVM+D6omotx7UWxE196DmIurEgzoRUXblu7hOwAyw+sd3saAnngHsI1eXCLyCLdh1bsMajWD+HIIX+JBa7sPfFsXeUqmTDKN53Ccxy/G0ZIkzqC3UGrTbqHCH4uuEVlgMknHP+zrGxyfMbSz0mmMrfFbs5FGRMQmnMyJ5hgUd9BYjWk/N6NyhljPy7rjWDH+7WPem1gy/Sxo/Iy+ea83wMy397ByytzW2fQ5sC1bTVGvf1pvS4PwL0zD1S7TrosXFtzrWcwbpnTakv6/fzP453ss21Vg/tt6MRu6MLy+NrwkNq+fc0XMzKug9sddralHjkUx03GvrTWVIaRedaDnsU9M3g30G+s2YejMah+BxbVPMvXDqTWfvtBiNrTej8Uhx3vOMPHlTb0ZjSM+sD1tvRgOzLV0d59t6U8uOGuDY2dabWvUJZYExB8RznlusV5SRnzTX1EbkH9Rna1yff3Ufw5zNsyJGqKdkfdtqOr1iL6uXyPgLMVi1WUM50L+YOz5YmcZCbYrxFcswK+3vq3TsHo+aPwAtRrD6+QxAypknIKHJSaD1ToDiDTHqKo/M4DZFHM6S4yVUR7fORG/R8uWsUbntObVKcZkdrdVjh+x1TnNvSj7hAWlW0sNB5Ruuoihp6KCkIZleE9290+u1rP0NETddQkyLmdanEyE+SauPU31abzk6vqpPeWZQ+MzHzl/MNh9ra4MxT0q2CGWp4+n2M3kktw331evK5rj5s4jeKNqrE7IaIzqRysUo1GSL2Rtf0LOlfURncsiDafThPUaaylTxqRlm0TGfHpFFde2txBv1ZTJ0XM/J6hp7XI8eOuihB908xtmGHeMe1NoQMxzBUzsgyrlU6ColjWfql8XpaEpvsD6iT0oW0tBgexOXLGRdlP2iROUNoHE2cJQeTmOZjsF3VijJUb9PHhu7li3/VTq5NefbXZrj1bO5OhMzIK6bxDWiVcOnuvy0zIElWHg/2ST/tX6UyK8JR7ShEtdnDmfWy4RO/GOKYKfkGSe02qTVUe7t5qeWPzGcDpU5O8fT7JQsZET2L4L9KaU5GdGPe3fAnKCzRUjIRobYnVHh3fh8nZE4x6wfN1J8q8HOt5hs2Zz4G7ru6sppLnLEwPvA2dLcNjo5IF8wJq6Ztu52bdfvPoi09yTcWcIU7Vy5Rvw/pt/mx8yTtZUZgRrGN5BrW+d7HynFLKijLu3y9TbI9HWlvFLI8ExLbfc/K9OVkmQ7FHGhPLhbD4Bzn56ZF86SjOTOV/rwPlqXzUXK0yU94miPKYpnuz/UOzDKfZ12yTVacx2aJUOYBbMiijB9pSzyMt96XmXqYbTz/wt1q+uy1pBipGwGlzUk5fdjitZcKROY1Tx/X9Fq8ms9W+pVz2dCc3HsrOWvofWn8NvIbZ7D6PRKVuEmzQGmYJ+sRrglWukRxutmiZeZmYaWfbb87Jw0vdyW88TXbN1sjH3SmMohzZpTnbUw9fPQeOnQeBmowzadNVotmnZjiZ6LsUVbn1aG8mvCrd2A8lykLHtkBjUKkNKNpcKoDkSqcoxvUO9EWhsirS6sVvc0wF3zIUj/Wl9e3V8Xu3ukbpFv0ycPjOOXAa3SEflcprU+UmMKyPkTbV/d1d+hFuTeIwuKlPkeJ64YPnXqUzkrJP253tlSsvPWIph7S290H2NjO1T/9QpyTGsip3VpEJ9Qj1jL78oRLVmkdcfniCjz3yWfiv2O+pjZ7W3fSVTyJ2y8yavK8uJIYUL6lzJv+yvR674Tv0YUE861d90DWs3fMFJgjMkk+D3LnN4Q7nJ8ksAebY/s56qd4lO8iSPROkm9UL8PsDEc9dq57s4tM2Iztl9AT9S6feu+HjK/JJijxO88J3pd2tXG2kddLD2fj1ZX73Ll5zo9zJf4Wn3MqY8bWdgor4zpqE+DubBEzbgwJoRLs1E0kb+Z5E1k5tOpUMqmt6FczjSwjXlB8ZJ0DxQRPu/umteb+1gYR2+FXo+wLjVukShhNi7V+QHX0mJW6uLKPsStF2t3o8TZiap2CkPd3S2s/WYLGZP1S5SUs+HeruydUpQiZ2GYQl/xjd6q+NCl+SkU/B0pX3RoOIbkDlvg326pbbX7Hm5DvNZ1zmhG1IK2YLAUe3f1OMs96nX02qHu0g/hEM5jBLqWpB/RTtpUdqYsS+5SD6f/hqxApmJRetuz+RhcLvJIVjk1Gc+ILJs8mpEy38VpOhbDIWQkZS7hfPhcQxrFsTLfaWo2BkNdHkGZQxMe5h5D2Du3vZvzcjnV62uVSygP3gXMiYvB4clfdaxi+4VYqMx5I++fA1qH4xrqZrf4X8dh+FhOzXmFcsvpu2YvA94694t1Rhb94eZrxnILmc3VHMN5psXorLfk58d+X9ToTaXOaN4/ffRH7RwwvBaK86CydIx3Z5GVN5QKngv4ZEjVf9Q/LsjfRnhd0KiSowklc05RTc30kKmZb1z6Rmc+C5HJ0qmSqUzNxhEtuhG7rfbVLfjZLjzAprdD+buU/Bex/u/PDqD1mKyHyaJz5qBDbTFlP+wp2oCe7f3ZKonxLi/f7W1DC56FH1Ar3vO9R/3xrm+7NLbqb5DwWr+rUjUoRSTLp3t2XfVgBOWTN84Bme/5RnSXnrNYfPNsHHC2aO5PLUu0oE/kmwW9SnzPkbJPc3Wqz+rx5ABv2HeL/FCkfkVtXW3ncc+VOB9Wcj5c4pyTdsocTp3P6u9mVXHZdrgMitzZie6XUpxtz/Pqc6M7lVz4Dno9fliDHzpStkj7rygSzlR9Nm9eQ3OuZXJPWCfKZCJZDxhndov3XR/ZntTwOgkY/51K9B1H0j2QpUf574hO2DKil2jd7JL0fNOxPpN6u0Za/T3K55fXbiz/XwarlaPN9d+t33iwufbZTf3fHHyofqJ+pq7BEv+N+gyIHaojYPAH9Vf1N/X3reHWH7f+tPVn7vrBBY35sSr92/rLfwFs2ay7</latexit>

Lp norms on
densities:✓<latexit sha1_base64="2JD/jKR1N5jqzrWaJC57kEeEWIc=">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D(↵, ) =<latexit sha1_base64="MRXohxEFGoI25S4PMP017weYREo=">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</latexit><latexit
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Z

d↵
( (x)
X dx

d
(x))p dx
dx

! defined only if ↵ ⌧ dx and

◆1/p

d↵
=
dx

d
dx Lp (dx)

⌧ dx.

↵

↵

↵

↵

Metrizes the strong topology.
weak

↵ !↵
D(↵, ↵ ) ⇡ cst

Wp (↵, ↵ ) =linear. Any entropy function Ï induces a Ï-divergence (also known
Csiszargence [Ciszár, 1967,
Ali and Divergence
Silvey, 1966] or f -divergence) as follows

Definition 8.2d↵(Ï-Divergences).
Let Ï d↵be an entropy function. For –dd–‹ be the Lebesgue
1 $Comparing$—+–decompositionof –1 with respect to —. T
d—
dx
dx
d
Imre
is defined by
Csiszár
3
4
⁄
d–
def.
Csiszár '-divergence: DÏ (–|—) = Ï
d— + ÏÕŒ –‹ (X )d—Xif –, — arenonnegativeand' convex,'(1)= 0, ' >0 Œ otherwise.
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Important if ↵(X ) 6= (X ).

In definition (8.1), the additional term ÏÕŒ –‹ (X ) is important t
Proposition:
D' > 0functional
is convex,) =topology
0 , ↵ =of measures.
' (↵|weak
defines a continuous(forDthelinear growth at infinity, as this is for instance the case for the abdefining the TV norm. If Ï as a super-linear growth, e.g. the usual enÏÕŒ = +Œ so that DÏ (–|—) = +Œ if – does not have a density withIn the discrete setting, assuming<latexit sha1_base64="J+PPDfpgiBmIu3T2fl+BnMjXkBg=">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ÿai ”xi

and

—=

i

ÿ

bi ”xii

are supported on the same set of n points (xi )ni=1 µ X , (8.1) define
n

ÿ

3

a

4

ÿ

linear. Any entropy function Ï induces a Ï-divergence (also known
Csiszargence [Ciszár, 1967,
Ali and Divergence
Silvey, 1966] or f -divergence) as follows

Definition 8.2d↵(Ï-Divergences).
Let Ï d↵be an entropy function. For –dd–‹ be the Lebesgue
1 $Comparing$—+–decompositionof –1 with respect to —. T
d—
dx
dx
d
Imre
is defined by
Csiszár
3
4
⁄
d–
def.
Csiszár '-divergence: DÏ (–|—) = Ï
d— + ÏÕŒ –‹ (X )d—Xif –, — arenonnegativeand' convex,'(1)= 0, ' >0 Œ otherwise.
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Important if ↵(X ) 6= (X ).

In definition (8.1), the additional term ÏÕŒ –‹ (X ) is important t
Proposition:
D' > Statistical
0functional
is convex,
D' (↵|weak) =topology
0 , ↵ =of measures.
defines a continuous
Divergences(for the
2is for instance2 the case for the ablinear growth at infinity, as
this
|s 1|
KL
defining the TV norm. If Ï as a super-linear growth, e.g. the usual en
|s 1|
TVÕ
TVhave
norm
Ï
=
+Œ
so
that
D
(–|—)
=
+Œ
if
–
does
not
a density withÏHellingerŒlog(s) s + 1 Generalized KL
χ 2 In the discrete setting, s
assuming
p<latexit sha1_base64="J+PPDfpgiBmIu3T2fl+BnMjXkBg=">AABB53ictVxfcxPJER8u/w7yj0se87KJ4QpSnGOcq0pSV6k6YxvwYcAgGbg7AbWS1kKw0gqtZAw6f4M85CmV13yFvCZV+Rz5BslTvkL6z8zOrDS7PesQpmzPjubX3dM709PdM6I7SYf5bGPjn+c++Na3v/Pd7314/sL3f/DDH/344kc/eZRn82kvOexlaTZ90o3zJB2Ok8PZcJYmTybTJB510+Rx99U2fv74OJnmw2zcnr2dJE9H8WA8PBr24hk0Pb/48d5okk1n8XgWDY+iS504vdJ5cnI16oyT11Gnm9DTpfXnF9c21jfoX7Raua4ra0r/O8g+ig5VR/VVpnpqrkYqUWM1g3qqYpVD+VpdVxtqAm1P1QLaplAb0ueJOlUXADuHXgn0iKH1FfwewNPXunUMz0gzJ3QPuKTwMwVkpC4DJoN+U6gjt4g+nxNlbK2ivSCaKNtb+NvVtEbQOlMvoFXCmZ6hOBzLTB2p39IYhjCmCbXg6Hqaypy0gpJHzqhmQGECbVjvw+dTqPcIafQcESansaNuY/r8X9QTW/G5p/vO1b9JystQItXSo88KCrE6JvoRvc05fMbypMB5ABQSPUasvSFdj2j0Y+i/gPZ7UE6pZnTShbKg1tNa5DYUH3JbRN6C4kPeEpH7UHzIfRF5AMWHPNBIxE5J5358C4oP3xI5P4DiQz4QkQ+h+JAPReQjKD7kIxH5FRQf8isReROKD3lTRN6B4kPeEZFtKD5kW0QeQvEhD0XkLhQfclcjq1fqFEpGdIbCqtyCepkHWooUWrZE+W6QdfRhbwSs6V4FVl7VO/DXj90J0GlSgd0NmHdHFVh55t0CG+nHyrboNu0mPuxtEbsHM8CP3ROxX6iXFdgvAlbaqwqsvNb2oZ8fK1vfu/Dkx94Vsfeg5sfKe9R9aPFj7wfsGJMK7IGIfaBeV2BDrP60Aivb/RbYFT9W3qfa0N+PDbGm8wqsbE8fgQfjx8q71WNo9WMfi9gn6qQC+0TEfgnW3Y/9MmCHfVeBNXvsBdpBBuSPJLBi66jFxarE2gSoxQL/tNhbUvKNu9AuYQYFZkCYkYi4VSBuBSL2C8R+sFx5YUdz8ndlLq0C0QpEdIu9CWszsX+/6I+1NACxUyB2lhB1Him+azOWY/IuTIuEnBU7F9ZCxpQV9htriZ4P9ZbXIO6XEDy3X9DMv0bREkZQqKk6ai+KPZ6RET3XId5Q9GZGaXjIuFlhFVzUiYjqelBdEfXWg3orouYe1FxEHXtQxyLKrnwX1wmYAVb/+C4W9MQzgH3k6hKBV7AFu85tWKMRzJ8D8AIfUst9+Nui2FsqdZJhNI/7JGY5npYs8RRqC7UG7TYq3KH4OqUVloBk3PO+jvHxCXMbC73m2AqfFjt5VGRMwukMSZ5BQQe9xYjWUzM6d6jllLw7rjXD3y7Wvak1w++Sxk/Ji+daM/xMSz87g+xtjW2fAduC1TTR2rf1pjQ4/8I0TP0C7bpocfGtjvScQXonDenv6Tezd4b3sk011o+tN6ORO+PLS+NrQsPqOXf03IwKek/s9Zpa1HgkYx332npTGTLaRcdaDvvU9M1gn75+M6bejMYBeFzbFHMvnHrT2TspRmPrzWg8Upz3PCVP3tSb0RjQM+vD1pvRwGxLrON8W29q2VEDHDvbelOrPqYsMOaAeM5zi/WKpuQnzTW1IfkH9dka1+df3ccwZ/OsiBHqKVnftppOt9jL6iUy/kICVm3WUA70L+aOD1amsVCbYnzFMsxK+/sqHbvHo+b3QYsRrH4+A5By5ilIaHISaL1ToHhdjLrKIzO4TRGHs+RoCdXRrTPRW7R8OWtUbntOrVJcZkdr9dghe53T3JuQT7hPmpX0sF/5hqsoShraL2lIptdEd+/0ei1rf0PETZYQk2Km9ehEiE/S6uNUn9Zbjo4v61OeGRQ+87HzF7PNR9raYMyTkS1CWep4uv1MHsltw331mrI5bv4sojeK9uqYrMaQTqRyMQo12WL2xhf0bGkf0pkc8mAaPXiPkaYyUXxqhll0zKdHZFFdeyvxRn2ZDB3Xc7K6xh7XowcOeuBBN49xtmHHuAe1NsQMh/DUDohyLhS6ykjjU/VJcTqa0Rusj+jTkoU0NNjeJCULWRdlvyhReQNonA0cpYfTWKZj8J0VSnLU75PHxq5ly3+ZTm7N+XZMc7x6NldnYvrEdZO4RrRq+FSXn5Y5sAQL7yeb5L/WjxL5NeGINlTi+szhzHoZ04l/QhHshDzjlFabtDrKvd381PInhtOBMmfneJqdkYWMyP5FsD9lNCcj+nHvDpgTdLYIKdnIELszLLwbn68zFOeY9eOGim812PmWkC2bE39D111dOc1Fjhh4HzhdmttGJ/vkCybEdaqtu13b9bsPIu09CXeWMEU7V64Q/6v02/yYebK2MiNQw/gGcm3rfO8jo5gFdRTTLl9vg0xfV8pLhQzPtNR2/7MyXSpJtkMRF8qDu3UfOPfomXnhLJmS3PlKH95H67K5SHmypEcc7RFF8Wz3B3oHRrmv0S65RmuuQ7NkALNgVkQRpq+URV7mW8+rTD2Mdv5/oW51XdYaUoyUzeCyhqT8fkLRmitlCrOa5+8rWk1+rU+XetXzGdNcHDlr+Rto/Tn8NnKb5zA63ZJVuEFzgCnYJ6sRbolWeoTxulHiZWamoWWfLT87J00vt+Us8TVbNxtjHzemckCz5kRnLUz9LDReOjReBuqwTWeNVoum3Vii52Js0danlaH8mnBrN6A8FynLHplBDQOkdGOpMKp9kaoc4xvUO5HWhkgrhtXqnga4az4E6V/ry6v7m2J3j9RN8m165IFx/NKnVTokn8u01kdqTAE5f6rtq7v6O9SC3LtkQZEy3+PEFcOnTj0qp4WkH+udLSM7by2Cubf0RvcxNrZD9V+vIEe0JnJalwbxKfVItPyuHNGSRVp3fI6IMv8x+VTsd9THzG5v+06ikj9h401eVZYXRwpj0r+UedtbiV73nPg1ophwrr3rLtBq/oaRAmNMJsHvWeb0hnCX45ME9mi7ZD9X7RSf4o0didZJ6oX6fYCN4ajXznV3bpkRm7H9Enqi1u1b9/WQ+aXBHCV+ZznRi2lXG2kfdbH0fDZasd7lys91epgv8bX6mFMfN7KwUV4Z01GfBXNhiZpxYUwIl2ajaCJ/M
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4

2

0

2
|– =s ÿ1|
a”

i xi

0

1

2

3

i
s log(s)ÿ distance
Hellingerand — =
bi ”xiKL i

are supported on the same set of n points (xir)ni=1 µrX , (8.1)
define2

Figure 8.1: Example of entropy functionals.

d↵ dd↵||↵: nbi = 0}.||TV =dH (↵, 3)= 4= {i œ JnK
1dxdx
dx L (dx)f the following Proposition can be found in [Liero et al., 2018, Thm 2.7]. ÿaded2 (dx)dx Lÿ

8.1. Ï-Divergences

Equivalence and non-equivalence1

Ï-Divergences

KL

KL dH

KL/2

TV

TV

)

KL
2

2

2/

TV
dH
2TV

2TV

2

W1

TV
dmax TVdH

22/

log(1 +

22

dH

KL/2

KL KdLH)

2

TV

dmax TV
W1

KL

2

dH

W1

W1

log(1 +d
TV TV dHTV
H
d
TV W /d
TV d

W1 /dmin

1

min

H

W1

W1

TV

H

Figure 8.2: Diagram of relation
8.2: Diagram of relationship between divergences(inspiredbywithGibbs
and Sudistance[2002]).
metricspace
groundÕrelationship between divergences (inspired by
Gibbs and Su [2002]). For X

space with ground distance d, dmax = sup(x,xÕ ) d(x, x ) def.
is the diameter of X . When X is d.

ÕÕÕ. d(x,distanced,ddiameter
of
X
When
d
=
min
x
).X is discreÕmax = sup(x,xÕ ) d(x, x ) is the
min
x”
=
x
Õ d(x, x ).
min
x”=x

2 (Hyperbolic geometry of KL). It is interesting to contrast the geometry of
ck-Leibler divergence to that defined by quadratic optimal transport when
Gaussians. As detailed for instance by Costa et al. [2015], the Kullbackergence has a closed form for Gaussian densities. In the univariate case,= N (m= N (m– , ‡–2 ) and — = N (m— , ‡—2 ), one has ↵ = N (m↵ , ↵ ),OT vs. KL (Fisher-Rao)

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1
KL(–|—) =
2

A

‡–2
+ log
‡—2

A 2B
‡
—
‡–2

,

)

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+
≠1 .
‡—2

(8.6) ) = |m↵
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Optimale infinitely far from all other Gaussians in the KL geometry. That geometry(hyperbolic)(Euclidean)ul when one wants to avoid dealing with singular covariances. To simplifys, one can look at the infinitesimal geometry of KL, which is obtained bya Taylor expansion at order 2

1
L(N (m + ”m , (‡ + ”‡ ) )|N (m, ‡ )) = 2
‡
2

2

3

4

1 2
2
”m + ”‡2 + o(”m, ”‡2 ).
2

Ô
Riemannian metric, the so-called Fisher metric, expressed over (m/ 2, ‡) œmatches exactly that of the hyperbolic Poincaré half-plane.
m Geodesics overre half circles centered along the ‡ = 0 line, and have an exponential speed,
nly reach the limit ‡ = 0 after an infinite time. Note in particular that if
t m– ”= m– , then the geodesic between (–, —) over this hyperbolic half planeve a constant standard deviation.
hyperbolic geometry over the space of Gaussian parameters (m, ‡) should bewith the Euclidean geometry associated to OT as described in Remark 2.31,
univariate caseW 22 (–, —) = |m– ≠ m— |2 + |‡– ≠ ‡— |2 .

(8.7)

shows a visual comparison of these two geometries and their respectivemOverview• Csiszar Divergences• Dual Norms and MMD• Minimum Kantorovitch Estimators• Deep Generative Models FittingDual Norms||↵

Dual norms: (aka
⇢Z Integral Probability Metrics)def.

||B = max

f (x)(d↵(x)Xd (x)) ; f 2 B

Dual Norms||↵

Dual norms: (aka
⇢Z Integral Probability Metrics)def.

||B = max

f (x)(d↵(x)Xd (x)) ; f 2 B

TV: B = {f ; ||f ||1 6 1}.
<latexit sha1_base64="TDxlt+jILqrlA1wV/WIktj9ztIw=">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</latexit><latexit
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Wasserstein 1: B = {f ; ||rf ||1 6 1}.
Flat norm: B = {f ; ||f ||1 6 1, ||rf ||1 6 1}.

Negative Sobolev: B = f ; k = 0, . . . , s, ||@ k f ||L2 (Rd ) 6 1<latexit sha1_base64="kYsZ7XdV6BNlr0+NiD51oaEozBg=">AAA92XictVtLcxu5EYY3r7Xz8ibHVKUmkZ3ybjkuSdmqJLulqrUkP7Tm2rJJ2d5dWio+hvRYQw7NIeUHl5ecUrnmB+WSP5F/kOSSQ/ID0t0ABhgSMw0ojmFJGBBfd6MB9AMYdidpks82N/924b1vffs73/3e+xcvff8HP/zRjy9/8JPHeTaf9uKjXpZm06fdTh6nyTg+miWzNH46mcadUTeNn3RP9/DzJ2fxNE+ycWv2ZhI/G3WG42SQ9DozaDq5fHw/HkL1LI6aWTdL47NPoiu70U7Ujsd5Lxv3F4Pl4nRn83o77Wez/HoE/9ufRO1xNh0t2pPOdJZ00uPTaLA8WTSOt6+1Hz067n+4jNpp/DLaWl45ubyxeWOT/kXrlS1V2RDq32H2wc+3RVv0RSZ6Yi5GIhZjMYN6Kjoih/K12BKbYgJtz8QC2qZQS+jzWCzFJcDOoVcMPTrQegq/h/D0tWodwzPSzAndAy4p/EwBGYmrgMmg3xTqyC2iz+dEGVuraC+IJsr2Bv52Fa0RtM7Ec2jlcLqnLw7HMhMD8TsaQwJjmlALjq6nqMxJKyh5ZI1qBhQm0Ib1Pnw+hXqPkFrPEWFyGjvqtkOf/516Yis+91TfufgHSXkVSiSaavRZQaEjzoh+RLM5h8+kPClwHgKFWI0Ra69I1yMa/Rj6L6D9PpQl1bROulAW1LqsRe5BcSH3WOQdKC7kHRbZgOJCNljkIRQX8lAhETslnbvxTSgufJPl/BCKC/mQRT6C4kI+YpGPobiQj1nkV1BcyK9Y5G0oLuRtFnkPigt5j0W2oLiQLRZ5BMWFPGKRt6C4kLcUsnqnTqFkRCdhduVNqJd5oKVIoeUmK98uWUcXdtdjT/cqsPyu3oe/buy+h07jCuwtj3U3qMDyK+8O2Eg3lrdFd8mbuLB3WewBrAA39oDFfi5eVGA/99hppxVYfq81oJ8by1vfL+DJjf2Cxd6HmhvL+6gH0OLGPvDwGJMK7CGLfSheVmB9rP60Asvb/SbYFTeW91Mt6O/G+ljTeQWWt6ePIYJxY3lv9QRa3dgnLPapeF2BfcpivwTr7sZ+6eFh31ZgtY+9RB5kSPFIDDu2jlqn2JVYmwC1DsM/LXxLSrFxF9o5zLDADAkzYhF3CsQdT0SjQDS85coLO5pTvMtzaRaIpieiW/gmrM3Y/v2iP9ZSD8R+gdhfQdRFpDjXeixnFF3oFg45KzwX1nzGlBX2G2uxWg/1llcjHpQQcm0/p5V/nbIlzKBQU3XUnhc+XiIjeq5DvKLsTY9S8+Bxs8Iq2KjXLKrrQHVZ1BsH6g2LmjtQcxZ15kCdsSiz821c22MFGP3jXCzoSa4AGSNXlwiigpvgde7CHo1g/RxCFPiIWh7A3ybl3lypkwyzefSTeMrxrGSJp1BbiA1oN1nhPuXXKe2wGCSTPR+oHB+f8GxjofactMLLwpNHxYmJP52E5BkWdDBajGg/hdG5Ry1Liu5kLQx/t9j3uhaGv0UaX1IUL2th+JmSfnYO2VsK2zoHtgm7aaK0b+qhNOT5i6Sh65fI66LFxVkdqTWD9F4H0j9QM3NwjnnZo5rUj6mH0cit8eWl8YXQMHrOLT2HUcHoSUa9uhYFj2Ss8l5TD5UhIy86VnKYp9CZwT59NTO6HkbjECKuPcq5F1Y9dPVOitGYehiNx0Keey4pktf1MBpDepb6MPUwGnja0lF5vqmHWnbUgMydTT3Uqo/pFBjPgOSaly0mKppSnDRX1BKKD+pPa+yYf92P4ZnNcZEj1FMysW01nW7hy+ol0vFCDFZtFigHxhdzKwYr01iIbTa/kjLMSv59nY7x8aj5Bmgxgt0v7wC4M/MUJNRnEmi9U6C4xWZd5ZFp3DaLw1UyWEG1VeuMjRYNX3lqVG47oVYuLzOjNXpsk73Oae1NKCZskGY5PTQqZ7iKIqehRklDPL0Q3b1V+7Ws/U0WN1lBTIqV1qMbIXmTVp+nurTetHR8Vd3yzKDIOx+zfvG0eaCsDeY8GdkilKWOp91PnyPZbehXrwtzxi0/i2hG0V6dkdVI6EYqZ7NQfVoso/EFPRvaR3QnhzwkjR7MY6SoTIS8NcNTdDxPj8ii2vaW44360id0sp6T1dX2uB49tNBDBzo8x9kDj3Efai3IGY7gqeWR5VwqdJWRxqfi18XtaEYzWJ/RpyULqWlIexOXLGRdlv28ROUVoHE1yCzdn8YqHY1vr1His36XPCZ3LVv+q3Rzq++3O7TGq1dz9UlMn7huE9eIdo281ZVPqxykBAvnJ9sUv9aPEvmFcEQbynE9tjhLvYzpxj+mDHZCkXFKu43bHeXe9vnU6iea06HQd+d4m52RhYzI/kXgnzJakxH92O8O6Bt0aRFSspE+dicpohtXrJOwa8zEcYmQbzWY9RaTLZsTf03X3l05rUWZMUg/sFxZ21onDYoFY+I6Vdbd7O1674NI856EvUokRbNWrhH/D+m3/tHrZGNtRaCGcQZyZetc85FRzoI66pCXr7dBuq8t5ZVChmMltfF/RqYrJcn2KeNCedBb94Fzj54lL1wlU5I7X+sj/WjdaS5SnqzoEUc7oCxe2v2h8sAo93Xykhu059q0SoawCmZFFqH7cqfIq3zreZWp+9HO/y/Uja7LWkOKkTAnuFJD3Pl+TNmaLWUKq1qu31PaTW6tT1d61fMZ01ocWXv5G2j9BfzWcutnPzrdklXYpTUgKZgnoxHZEq318OO1W+KlV6amZZ4NP7MmdS+75Tz5tbRuJsc+C6ZySKvmtTq10PXz0Hhh0XjhqcMW3TUaLep2bYlO2NyipW4rffmFcGsFUJ6zlPmITKMSDyntXMqPap+lyuf4GvWWpbXJ0urAbrVvA+w974N07/XV3f1N4d0jcZtimx5FYDJ/6dMuTSjm0q31mZqkgJw/VvbV3v1takHuXbKgSFm+x4k7Rt469agsC0l/pTxbRnbeWAT93tIr1Ufb2DbVf7OGHNGeyGlfasTH1CNW8ttyRCsW6YYVc0R08t+hmErGHfU5s93bzElUiidMvil3leElM4Ux6Z87eTtYy14PrPw1opxwrqLrLtAKn2GkIDH6JMEdWeY0Q+jl5E2CjGi7ZD/X7ZS8xRtbEt0gqRdix8PGyKzXrHV7bekR67F9BD1R62bWXT14fqk3R47feW70OuTVRipGXaw8n49WR3m58nOdHuYrfI0+5tTHzixMllfGtMWn3lykRGFcJMaHS9goQuQPkzxEZnk75UtZ99aUyycN0sY8p3yJew8UEa7o7pozmvuQGUd3jV6XsDY12cJRwtO4TJ0P2JYWT6Uurvkh2Xqx1hullieq8hSauu0tjP2WFjIm65cK7sxG9rZlb5eyFP4URlLoCflGb1V+aNP8FAr+joQrO9Qcfc4OmxDf3hR74tY7eBviparLE82IWtAW9Fdy744aZ7lHvY5eWtRt+j4c/HkkoGtO+oQ8aajskjIvuU3dn/4rsgJTEbPSm57hY7C58CNZ5xQynoQsGz+aROjv4oSORXPwGUmZiz8fea/BjWIg9HeawsagqfMjKHMI4aHfY/Cbc9M7nJfNqV5f61x8eUgvoG9cNA5v/qpzFdPPx0JNrRl59xzQOgxqqGtv8b+OQ/MxnMJ5+XLL6btmLzxmXfaL1YksxsPhe8Zw81nN1Rz9eWbF6Ey05OYn474oaKYyazTvnj7Go2YNaF4LIc9Beekk3l5FRl5fKngv4JIhE/8Sf7nAfxvhZUGjSo4QSvqeopqa7sFT09+4dI1Of+Yjk6FTJVOZmskjmvRG7J44ELfhZ6+IAEPfDpXfpZR/Eev+/mwfWgdkPfQpujw5aFNbTKcf5hatT8/qjPHk8sbW6reQ1ytH2zd+f2Pr4fbGZ7vqC8rvi5+JX0JasiV+Kz4Td2G4RyDSX8U/xb/Ff3baO3/Y+ePOn2TX9y4ozE9F6d/On/8Lryv3ow==</latexit><latexit
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Dual Norms||↵

Dual norms: (aka
⇢Z Integral Probability Metrics)def.

||B = max

f (x)(d↵(x)Xd (x)) ; f 2 B

TV: B = {f ; ||f ||1 6 1}.
<latexit sha1_base64="TDxlt+jILqrlA1wV/WIktj9ztIw=">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</latexit><latexit
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Wasserstein 1: B = {f ; ||rf ||1 6 1}.
Flat norm: B = {f ; ||f ||1 6 1, ||rf ||1 6 1}.

Negative Sobolev: B = f ; k = 0, . . . , s, ||@ k f ||L2 (Rd ) 6 1<latexit sha1_base64="kYsZ7XdV6BNlr0+NiD51oaEozBg=">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</latexit><latexit
sha1_base64="kYsZ7XdV6BNlr0+NiD51oaEozBg=">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</latexit>

Proposition: If span(B) is dense in C(X ),weak||↵||B ! 0 () ↵ ! 0<latexit sha1_base64="4ZAeYk66H5UMLGS9WVWPtnKxxI0=">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
sha1_base64="4ZAeYk66H5UMLGS9WVWPtnKxxI0=">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</latexit>

Dual Norms||↵

Dual norms: (aka
⇢Z Integral Probability Metrics)def.

||B = max

f (x)(d↵(x)Xd (x)) ; f 2 B

TV: B = {f ; ||f ||1 6 1}.
<latexit sha1_base64="TDxlt+jILqrlA1wV/WIktj9ztIw=">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</latexit><latexit
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Wasserstein 1: B = {f ; ||rf ||1 6 1}.
Flat norm: B = {f ; ||f ||1 6 1, ||rf ||1 6 1}.

Negative Sobolev: B = f ; k = 0, . . . , s, ||@ k f ||L2 (Rd ) 6 1<latexit sha1_base64="kYsZ7XdV6BNlr0+NiD51oaEozBg=">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</latexit><latexit
sha1_base64="kYsZ7XdV6BNlr0+NiD51oaEozBg=">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</latexit>

Proposition: If span(B) is dense in C(X ),weak||↵||B ! 0 () ↵ ! 0|| x<latexit sha1_base64="sGCKVodYxHhdA0I3rLjYlk4xZk4=">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
sha1_base64="sGCKVodYxHhdA0I3rLjYlk4xZk4=">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</latexit>

y ||TV = 2 if x 6= y

f 2 B needs to regular(e.g. s > d/2)

Hilbertian Norms on Measures
In X = Rd , smoothing with convolution:
def. R⇢↵ (x) = Rd h(x y)d↵(y)↵ ?h ↵ ? h = ⇢↵ dx
P↵ = i ai xi
⇢↵
↵
?h P
h
?
=
⇢↵ = i ai h(· xi )<latexit sha1_base64="g1cXtTwONdwX8llvbXgpPLmVxYA=">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</latexit><latexit
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In X = Rd , smoothing with convolution:
def. R⇢↵ (x) = Rd h(x y)d↵(y)↵ ?h ↵ ? h = ⇢↵ dx
P↵ = i ai xi
⇢↵
↵
?h P
h
?
=
⇢↵ = i ai h(· xi )<latexit sha1_base64="g1cXtTwONdwX8llvbXgpPLmVxYA=">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</latexit><latexit
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Hilbertian norm:||↵

2 def.
||k = ||⇢↵

⇢ ||2L2 (dx)

Hilbertian Norms on Measures
In X = Rd , smoothing with convolution:
def. R⇢↵ (x) = Rd h(x y)d↵(y)↵ ?h ↵ ? h = ⇢↵ dx
P↵ = i ai xi
⇢↵
↵
?h P
h
?
=
⇢↵ = i ai h(· xi )<latexit sha1_base64="g1cXtTwONdwX8llvbXgpPLmVxYA=">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</latexit><latexit
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2 def.
||k = ||⇢↵

Hilbertian norm:||↵

Kernel expression:||⇠||2k =

R

Rd

R R

R

Rd

⇢ ||2L2 (dx)h(xy)d⇠(y)

2

dx

0
0
k(y
y
)d⇠(y
)d⇠(y)
Rd Rd
def. R
Correlation kernel: k(y) = Rd h(x y)h(x)dx=

Comparison of Kernels

2. Integral Probability Metrics

123

h(x)

h ? (↵)

ED(R2=, Î·Î)||y||k(y)(–, —)(G,=.005)
.005

.02
(G,=
.02)

.05
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2|
gure 8.5: Top row: displayof space
Â such that Î– ≠ —Îk = ÎÂ ı (– ≠ —)ÎL2 (R|2|y|Sobolev) , formally defined overk(y) = e2 2urier as Â̂(Ê) =k̂0ú (Ê)
k )H where(Rnds for Gaussian kernel of variance ‡ . The kernel for ED(R , Î·Î) is Â(x) = 1/ ÎxÎ.
d+1
2(x, x ) = k0ú (x ≠ xÕ ). Bottom row: display of Â ı (– ≠ —). (G,‡)
2
2

úd

Õ

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Maximum Mean Discrepancies||⇠||2k =<latexit sha1_base64="Jq58dDTmbQsvw6fKXDFQnnT+DXs=">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</latexit><latexit
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R R

k(y
Rd Rd

y 0 )d⇠(y 0 )d⇠(y)

Theorem: if k̂(!) > 0, || · ||k metrizes weak convergence.
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Maximum Mean Discrepancies||⇠||2k =<latexit sha1_base64="Jq58dDTmbQsvw6fKXDFQnnT+DXs=">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</latexit><latexit
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R R

k(y
Rd Rd

y 0 )d⇠(y 0 )d⇠(y)

Theorem: if k̂(!) > 0, || · ||k metrizes weak convergence.
! Extends to general X using positive kernels (MMD).
R
2 def.
||⇠||k = X ⇥X k(x, y)d⇠(x)d⇠(y)MMD:= E(k(X, Y )), (X, Y ) ⇠ ⇠ indep.
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Maximum Mean Discrepancies||⇠||2k =<latexit sha1_base64="Jq58dDTmbQsvw6fKXDFQnnT+DXs=">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</latexit><latexit
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R Ry 0 )d⇠(y 0 )d⇠(y)

k(y
Rd Rd

Theorem: if k̂(!) > 0, || · ||k metrizes weak convergence.
! Extends to general X using positive kernels (MMD).
R
2 def.
||⇠||k = X ⇥X k(x, y)d⇠(x)d⇠(y)MMD:= E(k(X, Y )), (X, Y ) ⇠ ⇠ indep.
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||↵

2

|| =

X
i,i0

ai ai0 k(xi , xi0 )2X

ai bj k(xi , yj ) +

j,j 0

i,j

xi

X

yj

bj bj 0 k(yj , yj 0 )Comparison of Dual Norms

↵

↵
t

3
2
1
0
-3

t/2

3
Energy
Gauss
W1

2

Flat

-2

1

-1

0

1

2

3

0
-3

Energy
Gauss
W1
Flat

-2

-1

0

1

2

3

Csiszar Divergence vs Dual Norms

Csiszár Zdivergences:
✓ ◆
def.

D' (↵| ) =

D'

'

X

d↵
d

d

↵

Strong topology! KL, TV,||↵2

, Hellinger . . .

def.

||B = max
f 2B

↵
X

Dual
Z norms:f (x)(d↵(x)d (x))

X|| · ||B

Weak topology

X! W1 , flat, RKHS⇤ , energy dist, . . .

OT Loss for Diffeomorphic Registration
Joint work with J. Feydy, B. Charier, F-X. Vialard.
Shape registration:

min D('(µ), ⌫) + R(')' di↵eo
regularity
loss

µ

⌫

OT Loss for Diffeomorphic Registration
Joint work with J. Feydy, B. Charier, F-X. Vialard.
Shape registration:

min D('(µ), ⌫) + R(')' di↵eo
regularity
loss

Hilbertian loss (MMD/RKHS):

D(µ, ⌫) = ||k ? (µ(µ⌫) ? k⌫)||2L2µ⌫OT Loss for Diffeomorphic Registration+ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ˙ƇƚȀǞɾʿ ŏɾ ŏ żȩɯɾ ȩ

Joint work with J. Feydy, B. Charier, F-X. Vialard.
Shape registration:

min D('(µ), ⌫) + R(')' di↵eo
regularity
loss

Hilbertian loss (MMD/RKHS):

D(µ, ⌫) = ||k ? (µ(µ⌫)||2L2µSinkhorn divergence:D(µ, ⌫) = W̄" (µ, ⌫)

it 0

⌫

+ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ˙ƇƚȀǞɾʿ ŏɾ ŏ żȩɯɾ ȩ

it 250_Ǟǃʕɟƚࠊ ëɾŏɟɾǞȘǃ ƚɯɾǞȒŏɾƚࠒ

⌫) ? k_Ǟǃʕɟƚࠊ +ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ɔȀŏȘࠒ

OT Loss for Diffeomorphic Registration+ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ˙ƇƚȀǞɾʿ ŏɾ ŏ żȩɯɾ ȩ

Joint work with J. Feydy, B. Charier, F-X. Vialard.
Shape registration:

min D('(µ), ⌫) + R(')' di↵eo
regularity
loss

Hilbertian loss (MMD/RKHS):

D(µ, ⌫) = ||k ? (µ⌫)||2L2+ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ˙ƇƚȀǞɾʿ ŏɾ ŏ żȩɯɾ ȩ

Sinkhorn divergence:D(µ, ⌫) = W̄" (µ, ⌫)it 250_Ǟǃʕɟƚࠊ ëɾŏɟɾǞȘǃ ƚɯɾǞȒŏɾƚࠒ

output⌫) ? koutput

input

it 0

⌫

input

(µ

µ

_Ǟǃʕɟƚࠊ +ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ɔȀŏȘࠒ

OT Loss for Diffeomorphic Registration+ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ˙ƇƚȀǞɾʿ ŏɾ ŏ żȩɯɾ ȩ

Joint work with J. Feydy, B. Charier, F-X. Vialard.
Shape registration:

min D('(µ), ⌫) + R(')' di↵eo
regularity
loss

Hilbertian loss (MMD/RKHS):

D(µ, ⌫) = ||k ? (µ⌫)||2L2+ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ˙ƇƚȀǞɾʿ ŏɾ ŏ żȩɯɾ ȩ

Sinkhorn divergence:D(µ, ⌫) = W̄" (µ, ⌫)it 250_Ǟǃʕɟƚࠊ ëɾŏɟɾǞȘǃ ƚɯɾǞȒŏɾƚࠒ

output⌫) ? k_Ǟǃʕɟƚࠊ +ȩȒɔʕɾǞȘǃ ɾǕƚ °÷ ɔȀŏȘࠒ

output

input

it 0

⌫

input

(µ

µ

! Do not use OT for registration . . . but as a loss.
! Sinkhorn’s iterates “propagate” a small bandwidth kernel.
! Automatic di↵erentation: game changer for advanced loss and models.

Overview• Csiszar Divergences• Dual Norms and MMD• Minimum Kantorovitch Estimators• Deep Generative Models Fitting

Density Fitting and Generative Models
Observations:

def.

=

1
n

Pn

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i=1 xiParametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
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Observations:

def.

=

1
n

Pn

↵✓
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Parametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
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Density fitting: d↵✓ (x) = ⇢✓ (x)dxc |↵✓ ) =min KL(

def.

✓

X
i

log(⇢✓ (xi ))Maximum
likelihood (MLE)xi

Density Fitting and Generative Models
Observations:

def.

=

1
n

↵✓

Pn

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Parametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
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Density fitting: d↵✓ (x) = ⇢✓ (x)dxc |↵✓ ) =min KL(

def.

✓

X

log(⇢✓ (xi ))Maximum
likelihood (MLE)i

Generative model fit: ↵✓ = g✓,] ⇣c |↵✓ ) = +1
KL(

! MLE undefined.
! Need a weaker metric.

g✓
↵✓

⇣
Z

X

xi

Loss Functions for Measures
Density fitting: min D(↵✓ , )✓Optimal Transport
DistancesR
def.
Wpp (↵, ) = mind(x, y)p d⇡(x, y)⇡2U (↵, )

= n1

P

i xi

⇡?
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</latexit>↵<latexit 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Loss Functions for Measures
Density fitting: min D(↵✓ , )✓= n1

Optimal Transport
DistancesR
def.
Wpp (↵, ) = mind(x, y)p d⇡(x, y)

P

i xi

Maximum Mean Discrepancy (MMD)R
2 def.
||↵||k = k(x, y)d(↵(x)(x))d(↵(y)(y))

Gaussian: k(x, y) = e.

Energy distance: k(x, y) =↵<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit><latexit<latexit
sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit>⇡2U (↵, )||x y||2
2 2

⇡?

||x

y||2 .

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Loss Functions for Measures= n1

Density fitting: min D(↵✓ , )✓Optimal Transport
DistancesR
def.
Wpp (↵, ) = mind(x, y)p d⇡(x, y)

P

i xi

↵
<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit><latexit<latexit
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2 def.
||↵||k = k(x, y)d(↵(x)(x))d(↵(y)(y))

Gaussian: k(x, y) = e||x y||2
2 2.

Energy distance: k(x, y) =

||x

def.

p
W̄p,"
(↵,

min

⇡2U (↵, )def.
)p = Wpp," (↵, )p↵<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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<latexit
sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit>y||2 .

dp d⇡ + "KL(⇡|↵ ⌦ )p
1Wp," (↵,
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<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit><latexit<latexit
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2 def.
||↵||k = k(x, y)d(↵(x)(x))d(↵(y)(y))

Gaussian: k(x, y) = e||x y||2
2 2.

Energy distance: k(x, y) =

||x

def.

p
W̄p,"
(↵,

min

⇡2U (↵, )def.
)p = Wpp," (↵, )p↵<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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<latexit
sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit>y||2 .

dp d⇡ + "KL(⇡|↵ ⌦ )p
1Wp," (↵,
2

Theorem: [Genevay, P, C, 17]

)p

p
W̄",p
(↵,

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↵
<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit><latexit<latexit
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2 def.
||↵||k = k(x, y)d(↵(x)(x))d(↵(y)(y))

Gaussian: k(x, y) = e||x y||2
2 2.

Energy distance: k(x, y) =

||x

def.

p
W̄p,"
(↵,

min

⇡2U (↵, )def.
)p = Wpp," (↵, )p↵<latexit sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">AABB0HictVzdchPJFW42fwv5WTa5zM0kXlJsiiWGUJWktlK1xjbGiwCDZMPuCqiRNBaCkUZoJPOjdaVym1fIbfIceY68QXKVV8j56Z7ukXrm9DiELts9rf7OOX2m+/Q5p1v0pukon29u/vPcB9/57ve+/4MPz1/44Y9+/JOPLn7806M8W8z6yWE/S7PZ416cJ+lokhzOR/M0eTydJfG4lyaPei+38fNHJ8ksH2WTzvztNHkyjoeT0fGoH8+h6dnFj7pEYzlLBqdRN06fXdzYvLpJ/6L1yjVd2VD630H2cXSoumqgMtVXCzVWiZqoOdRTFascyjfqmtpUU2h7opbQNoPaiD5P1Km6ANgF9EqgRwytL+H3EJ6+0a0TeEaaOaH7wCWFnxkgI3UJMBn0m0EduUX0+YIoY2sV7SXRRNnewt+epjWG1rl6Dq0SzvQMxeFY5upY/Z7GMIIxTakFR9fXVBakFZQ8ckY1BwpTaMP6AD6fQb1PSKPniDA5jR11G9Pn/6Ke2IrPfd13of5NUl6CEqm2Hn1WUIjVCdGP6G0u4DOWJwXOQ6CQ6DFi7TXpekyjn0D/JbTfg3JKNaOTHpQltZ7WIreh+JDbInIPig+5JyJbUHzIlog8gOJDHmgkYmekcz++DcWHb4ucH0DxIR+IyIdQfMiHIvIIig95JCK/huJDfi0ib0HxIW+JyDtQfMg7IrIDxYfsiMhDKD7koYjcheJD7mpk9UqdQcmIzkhYlVtQL/NAS5FCy5Yo302yjj7szYA13a/Ayqt6B/76sTsBOk0qsLsB8+64AivPvD2wkX6sbItu027iw94WsfswA/zYfRH7pXpRgf0yYKW9rMDKa60F/fxY2frehSc/9q6IvQc1P1beo+5Dix97P2DHmFZgD0TsA/WqAhti9WcVWNnut8Gu+LHyPtWB/n5siDVdVGBle3oEHowfK+9Wj6DVj30kYh+rNxXYxyL2K7DufuxXATvsuwqs2WMv0A4yJH8kgRVbRy0uViXWpkAtFvinxd6Skm/cg3YJMywwQ8KMRcRegdgLRLQKRCtYrrywozn5uzKXdoFoByJ6xd6EtbnYf1D0x1oagNgpEDsriDqPFN+1GcsJeRemRULOi50LayFjygr7jbVEz4d6y2sQ90sIntvPaeZfoWgJIyjUVB2158Uez8iInusQryl6M6M0PGTcvLAKLuqNiOp5UD0R9daDeiuiFh7UQkSdeFAnIsqufBfXDZgBVv/4Lpb0xDOAfeTqEoFXsAW7zm1YoxHMnwPwAh9Sy33426bYWyp1kmE0j/skZjmelCzxDGpLtQHtNircofg6pRWWgGTc876O8fEJcxtLvebYCp8WO3lUZEzC6YxInmFBB73FiNZTMzp3qOWUvDuuNcPfLta9qTXD75LGT8mL51oz/FxLPz+D7B2N7ZwB24bVNNXat/WmNDj/wjRM/QLtumhx8a2O9ZxBem8a0t/Xb2b/DO9lm2qsH1tvRiN3xpeXxteEhtVz7ui5GRX0ntjrNbWo8UgmOu619aYyZLSLTrQc9qnpm8E+A/1mTL0ZjQPwuLYp5l469aazd1qMxtab0ThSnPc8JU/e1JvRGNIz68PWm9HAbEus43xbb2rZUQMcO9t6U6s+oSww5oB4znOL9Ypm5CctNLUR+Qf12RrX51/fxzBn87SIEeopWd+2mk6v2MvqJTL+QgJWbd5QDvQvFo4PVqaxVNfF+IplmJf293U6do9HzbdAixGsfj4DkHLmKUhochJovVOgeE2MusojM7jrIg5nyfEKqqtb56K3aPly1qjc9oxapbjMjtbqsUv2Oqe5NyWfsEWalfTQqnzDVRQlDbVKGpLpNdHdO71ey9rfFHHTFcS0mGl9OhHik7T6ONWn9baj40v6lGcOhc987PzFbPOxtjYY82Rki1CWOp5uP5NHcttwX72ibI6bP4vojaK9OiGrMaITqVyMQk22mL3xJT1b2od0Joc8mEYf3mOkqUwVn5phFh3z6RFZVNfeSrxRXyZDx/WcrK6xx/XooYMeetDNY5xt2DHuQa0DMcMhPHUCopwLha4y0vhMfVacjmb0Busj+rRkIQ0NtjdJyULWRdnPS1ReAxpnA0fp4TRW6Rh8d42SHPX75LGxa9nyX6KTW3O+HdMcr57N1ZmYAXG9TlwjWjV8qstPqxxYgqX3k+vkv9aPEvk14Yg2VOL61OHMepnQiX9CEeyUPOOUVpu0Osq93fzU6ieG04EyZ+d4mp2RhYzI/kWwP2U0JyP6ce8OmBN0tggp2cgQuzMqvBufrzMS55j140aKbzXY+ZaQLVsQf0PXXV05zUWOGHgfOF2Z20YnLfIFE+I609bdru363QeR9p6EO0uYop0rl4n/p/Tb/Jh5srE2I1DD+AZybet87yOjmAV1FNMuX2+DTF9Xyk8KGZ5qqe3+Z2X6pCTZDkVcKA/u1gPg3Kdn5oWzZEZy52t9eB+ty+Yi5emKHnG0xxTFs90f6h0Y5b5Cu+QGrbkuzZIhzIJ5EUWYvlIWeZVvPa8y9TDa+f+FutV1WWtIMVI2g8sakvL7CUVrrpQpzGqevy9pNfm1PlvpVc9nQnNx7Kzlb6H1F/DbyG2ew+j0SlbhJs0BpmCfrEa4JVrrEcbrZomXmZmGln22/OycNL3clrPE12zdbIx90pjKAc2aNzprYepnofHCofEiUIcdOmu0WjTtxhI9E2OLjj6tDOXXhFunAeWFSFn2yAxqFCClG0uFUR2IVOUY36DeibQ2RVoxrFb3NMBd8yFI/1pfXd3fFrt7pG6Rb9MnD4zjlwGt0hH5XKa1PlJjCsj5hrav7urvUgty75EFRcp8jxNXDJ869amcFpL+Su9sGdl5axHMvaXXuo+xsV2q/3YNOaY1kdO6NIgb1CPR8rtyRCsW6arjc0SU+Y/Jp2K/oz5mdnvbdxKV/Akbb/Kqsrw4UpiQ/qXM2/5a9LrvxK8RxYQL7V33gFbzN4wUGGMyCX7PMqc3hLscnySwR9sj+7lup/gUb+JIdJWkXqo/BtgYjnrtXHfnlhmxGduvoSdq3b51Xw+ZXxrMUeJ3lhO9mHa1sfZRlyvPZ6MV612u/Fynh8UKX6uPBfVxIwsb5ZUxXfV5MBeWqBkXxoRwaTaKJvI3k7yJzHw6FUrZ9DaUy5kGtjHPKV6S7oEiwufdXfZ6c58K4+it0esR1qXGLRIlzMZlOj/gWlrMSp1f24e49XztbpQ6O1HVTmGou7uFtd9sIROyfqmScjbc25W9W4pS5CwMU+grvtFbFR+6ND+Hgr8j5YsODceQ3GEb/Nstta1238NtiFe6zhnNiFrQFgxWYu9Yj7Pco15HrxzqLv0QDuE8RqBrSfoR7aRNZWfKsuQu9XD6r8kKzFQiSm97Nh+Dy0UeyTqnJuMZkWWTRzNS5rs4TcdiOISMpMwlnA+fa0ijOFbmO03NxmCoyyMoc2jCw9xjCHvntndzXi6nen2tcwnlwbuAOX<latexit
sha1_base64="ethir/69MZ9BTJ7/iqpADCSE0QY=">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</latexit>y||2 .

dp d⇡ + "KL(⇡|↵ ⌦ )p
1Wp," (↵,
2

Theorem: [Genevay, P, C, 17]

)p

p
W̄",p
(↵,

p
1
W
p," (↵,
2

⇡"?

↵

)p

"!0

! Wpp (↵, )) "!+1! ||↵||2k

for k(x, y) =– Scale free (no , no heavy tail kernel).

Best of both worlds:! cross-validate "<latexit sha1_base64="z84mRX++8b7jOXKAIdnn/tixKE0=">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</latexit><latexit
sha1_base64="z84mRX++8b7jOXKAIdnn/tixKE0=">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</latexit>kSinkhorn divergences R[Genevay, Peyré, Cuturi 17]
Wp",p (↵, ) =⇡?

– Non-Euclidean, arbitrary ground distance.

– Less biased gradient.
– No curse of dimension (low sample complexity).

d(x, y)pSample Complexity↵ˆlog(||↵
ˆ

d=2

↵

n

↵
ˆ 0 ||k )<latexit sha1_base64="tOLVHuCcODYvL/AduiUQnUlHORs=">AABB0XictVzdkhO5FRabv4X8scllbjqZJbWbImQgVCWprVTtMDMMsxgYsAfYXQPltnuMoe02btv8eKcqldu8Qm6T18hz5A2Sq7xCzo/UUtvqPuoJQTUzalnfOUenpaNzjmTiaTrK59vb/zz3wbe+/Z3vfu/D8xe+/4Mf/ujHFz/6ycM8W8z6yXE/S7PZ47iXJ+lokhzPR/M0eTydJb1xnCaP4pe7+PmjZTLLR9mkM387TZ6Me8PJ6GTU782h6dnFi12isYrTRXIadXvps4tb21e26V+0WbmqK1tK/zvKPoqOVVcNVKb6aqHGKlETNYd6qnoqh/K1uqq21RTanqgVtM2gNqLPE3WqLgB2Ab0S6NGD1pfwewhPX+vWCTwjzZzQfeCSws8MkJG6BJgM+s2gjtwi+nxBlLG1ivaKaKJsb+FvrGmNoXWunkOrhDM9Q3E4lrk6Ub+nMYxgTFNqwdH1NZUFaQUlj5xRzYHCFNqwPoDPZ1DvE9LoOSJMTmNH3fbo839RT2zF577uu1D/JikvQYlUW48+Kyj01JLoR/Q2F/AZy5MC5yFQSPQYsfaadD2m0U+g/wra70I5pZrRSQxlRa2ntchdKD7krog8gOJDHojIFhQfsiUij6D4kEcaidgZ6dyPb0Px4dsi5/tQfMj7IvIBFB/ygYh8CMWHfCgiv4LiQ34lIm9C8SFvisjbUHzI2yKyA8WH7IjIYyg+5LGI3IfiQ+5rZPVKnUHJiM5IWJU7UC/zQEuRQsuOKN8Nso4+7I2ANd2vwMqreg/++rF7ATpNKrD7AfPupAIrz7wDsJF+rGyLbtFu4sPeErGHMAP82EMR+4V6UYH9ImClvazAymutBf38WNn63oEnP/aOiL0LNT9W3qPuQYsfey9gx5hWYI9E7H31qgIbYvVnFVjZ7rfBrvix8j7Vgf5+bIg1XVRgZXv6EDwYP1berR5Bqx/7SMQ+Vm8qsI9F7Jdg3f3YLwN22HcVWLPHXqAdZEj+SAIrto5ar1iVWJsCtZ7APy32lpR84xjaJcywwAwJMxYRBwXiIBDRKhCtYLnywo7m5O/KXNoFoh2IiIu9CWtzsf+g6I+1NACxVyD21hB1Him+azOWJXkXpkVCzoudC2shY8oK+421RM+HestrEPdKCJ7bz2nmX6ZoCSMo1FQdtefFHs/IiJ7rEK8pejOjNDxk3LywCi7qjYiKPahYRL31oN6KqIUHtRBRSw9qKaLsyndx3YAZYPWP72JFTzwD2EeuLhF4BTuw69yCNRrB/DkCL/ABtdyDv22KvaVSJxlG87hPYpbjSckSz6C2UlvQbqPCPYqvU1phCUjGPe/pGB+fMLex0muOrfBpsZNHRcYknM6I5BkWdNBbjGg9NaNzm1pOybvjWjP8rWLdm1oz/D5p/JS8eK41w8+19PMzyN7R2M4ZsG1YTVOtfVtvSoPzL0zD1C/QrosWF9/qWM8ZpPemIf1D/WYOz/BedqnG+rH1ZjRyZ3x5aXxNaFg9546em1FB74m9XlOLGo9kouNeW28qQ0a76ETLYZ+avhnsM9BvxtSb0TgCj2uXYu6VU286e6fFaGy9GY2HivOep+TJm3ozGkN6Zn3YejMamG3p6Tjf1ptadtQAx8623tSqTygLjDkgnvPcYr2iGflJC01tRP5BfbbG9fk39zHM2TwtYoR6Sta3raYTF3tZvUTGX0jAqs0byoH+xcLxwco0VuqaGF+xDPPS/r5Jx+7xqPkWaDGC1c9nAFLOPAUJTU4CrXcKFK+KUVd5ZAZ3TcThLDlZQ3V161z0Fi1fzhqV255RqxSX2dFaPXbJXuc096bkE7ZIs5IeWpVvuIqipKFWSUMyvSa6e6fXa1n72yJuuoaYFjOtTydCfJJWH6f6tN52dHxJn/LMofCZj52/mG0+0dYGY56MbBHKUsfT7WfySG4b7quXlc1x82cRvVG0V0uyGiM6kcrFKNRki9kbX9GzpX1MZ3LIg2n04T1GmspU8akZZtExnx6RRXXtrcQb9WUydFzPyeoae1yPHjrooQfdPMbZhR3jLtQ6EDMcw1MnIMq5UOgqI43P1K+L09GM3mB9RJ+WLKShwfYmKVnIuij7eYnKa0DjbOAoPZzGOh2D725QkqN+nzw2di1b/kt0cmvOt3s0x6tnc3UmZkBcrxHXiFYNn+ry0zoHlmDl/eQa+a/1o0R+TTiiDZW4PnU4s14mdOKfUAQ7Jc84pdUmrY5ybzc/tf6J4XSkzNk5nmZnZCEjsn8R7E8ZzcmIfty7A+YEnS1CSjYyxO6MCu/G5+uMxDlm/biR4lsNdr4lZMsWxN/QdVdXTnORIwbeB07X5rbRSYt8wYS4zrR1t2u7fvdBpL0n4c4SpmjnyifE/1P6bX7MPNnamBGoYXwDubZ1vveRUcyCOurRLl9vg0xfV8qPCxmeaqnt/mdl+rgk2R5FXCgP7tYD4NynZ+aFs2RGcucbfXgfrcvmIuXpmh5xtCcUxbPdH+odGOW+TLvkFq25Ls2SIcyCeRFFmL5SFnmdbz2vMvUw2vn/hbrVdVlrSDFSNoPLGpLy+wlFa66UKcxqnr8vaTX5tT5b61XPZ0Jzceys5W+g9efw28htnsPoxCWrcIPmAFOwT1Yj3BJt9AjjdaPEy8xMQ8s+W3<latexit
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log(n)↵
ˆlog(W (ˆ↵, ↵ˆ 0 ))
d=3

Theorem: E(|W (ˆ↵, ˆ)
E(|||ˆ
↵ ˆ||kW (↵, )|) = O(n1d)

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log(W̄" (↵,ˆ ↵ˆ 0 ))

d=2log(n)

d=5log(n)

Sample Complexity↵ˆlog(||↵
ˆ

d=2

↵

n

↵
ˆ 0 ||k )<latexit sha1_base64="tOLVHuCcODYvL/AduiUQnUlHORs=">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<latexit
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log(n)↵
ˆlog(W (ˆ↵, ↵ˆ 0 ))
d=3

Theorem: E(|W (ˆ↵, ˆ)
E(|||ˆ
↵ ˆ||kW (↵, )|) = O(n1d)

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log(W̄" (↵,ˆ ↵ˆ 0 ))

Optimal transport: su↵ers from curse of dimensionality.

! Adapt to support dimensionality [Weed, Bach 2017]

d=2

log(n)

d=5log(n)

Open problem: sample complexity of W̄" ?

Overview• Csiszar Divergences• Dual Norms and MMD• Minimum Kantorovitch Estimators• Deep Generative Models Fitting

Deep Discriminative vs Generative Models

z

x

d⇠

Generative

Discriminatived⇠ (x) = ⇢(⇠K (. . . ⇢(⇠2 (⇢(⇠1 (x) . . .)Deep networks:g✓ (z) = ⇢(✓K (. . . ⇢(✓2 (⇢(✓1 (z) . . .)
⇢
⇠1 ⇢
✓
⇠2 ⇢
✓1 ⇢ 2zg✓x

Deep Discriminative vs Generative Models

Generative

Discriminatived⇠ (x) = ⇢(⇠K (. . . ⇢(⇠2 (⇢(⇠1 (x) . . .)Deep networks:g✓ (z) = ⇢(✓K (. . . ⇢(✓2 (⇢(✓1 (z) . . .)
⇢
⇠1 ⇢
✓
⇠2 ⇢
✓1 ⇢ 2

z

x

z

g✓

d⇠

x

z2

g✓

x

z
d⇠
Z

X

z1

Training Architecture
def.

↵✓

g✓
Z

min E(✓) = W̄" (↵✓ , )✓Stochastic gradient descent✓(`) = ✓(`)
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⌧` rÊL (✓)1
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X

y1
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⌧` rÊL (✓)1
Ê(✓) = W̄"L ( mPi g✓ (zi ),)Training Architecture
def.

x1 ↵✓

min E(✓) = W̄" (↵✓ , )✓<latexit sha1_base64="TKa8+ATXQYu13dSUNb6eZuTq9gM=">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
sha1_base64="TKa8+ATXQYu13dSUNb6eZuTq9gM=">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</latexit>

z1g✓✓(`) = ✓(`)

X

y1

e C/"

C

K

...

Generative model

Input data

...

...

⌧` rÊL (✓)1
Ê(✓) = W̄"L ( m(c(xi , yj ))i,j✓2(x1 , . . . , xm )(z1 , . . . , zm )

...

def.

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sha1_base64="7H1S8QwhorAsfW/ryppJL5U4Ekg=">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</latexit>

randn
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yn

(y1 , . . . , yn )zm

Z

Stochastic gradient descent

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Sinkhorn, ` = 0, . . . , L

1

Automatic Differentiation
Setup: E : Rn ! R computable in K operations.

pHypothesis: elementary operations (a ⇥ b, log(a), a . . . )and their derivatives cost O(1).
Question: What is the complexity of computing rE : Rn ! Rn ?

Automatic Differentiation
Setup: E : Rn ! R computable in K operations.

pHypothesis: elementary operations (a ⇥ b, log(a), a . . . )and their derivatives cost O(1).
Question: What is the complexity of computing rE : Rn ! Rn ?

Finite di↵erences:1
rE(✓) ⇡ (E(✓ + " 1 ) E(✓), . . . E(✓ + " 1 ) E(✓))"
K(n + 1) operations, intractable for large n.

Automatic Differentiation
Setup: E : Rn ! R computable in K operations.

pHypothesis: elementary operations (a ⇥ b, log(a), a . . . )and their derivatives cost O(1).
Question: What is the complexity of computing rE : Rn ! Rn ?

Finite di↵erences:1
rE(✓) ⇡ (E(✓ + " 1 ) E(✓), . . . E(✓ + " 1 ) E(✓))"
K(n + 1) operations, intractable for large n.

Theorem: there is an algorithm to compute rE in O(K) operations.
[Seppo Linnainmaa, 1970]
This algorithm is reverse mode
automatic di↵erentiation
Seppo
Linnainmaa

Computational Graph

Computational Graph

Computer program , directed acyclic graph , linear ordering of nodes (✓r )rforwardcomputing `function `(✓1 , . . . , ✓M )for r = M + 1, . . . , R✓r = gr (✓Parents(r) )return ✓R

g4

✓2
input
✓1

g3

✓3

✓4

outputg5✓5

Examplep`(✓1 , ✓2 ) = ✓2 e
✓ 1 + ✓ 2 e ✓1
def.

✓1
✓2
input

def.

g 3 ✓3 = e

✓1

g4

def.

✓1

g 5 ✓5 = ✓1 + ✓4

def.

✓4 = ✓2 ✓3

def.

g 6 ✓6 =

p

g7

✓5
def.

✓7 = ✓4 ✓6

output `Examplep`(✓1 , ✓2 ) = ✓2 e
✓ 1 + ✓ 2 e ✓1
def.

def.

✓1

g 3 ✓3 = e

✓1

g4

✓2

✓1

def.

g 5 ✓5 = ✓1 + ✓4def.

g 6 ✓6 =

p

g7

def.

✓4 = ✓2 ✓3

input

✓1

...

gj

✓i

“ @✓ = X@✓1

def.

✓7 = ✓4 ✓6

output `✓j = gj (✓i )i6j

gk

Chain rules:j✓5@✓j @✓i@✓i @✓1”i2Parent(j)@i gj (✓)

“Classical” evaluation: forward.
Complexity ⇠ #inputs.

✓k = gk (✓` )`6k

. . . ✓N

Examplep`(✓1 , ✓2 ) = ✓2 e
✓ 1 + ✓ 2 e ✓1
def.

def.

✓1

g 3 ✓3 = e

✓1

g4

✓2

✓1

def.

g 5 ✓5 = ✓1 + ✓4def.

g 6 ✓6 =

p

g7

def.

✓4 = ✓2 ✓3

✓5
def.

✓7 = ✓4 ✓6

input

✓1

...

output `gj✓i✓j = gj (✓i )i6j

gk

. . . ✓N✓k = gk (✓` )`6k

Chain rules:“ @✓ = Xj@✓1@✓j @✓i@✓i @✓1”i2Parent(j)@i gj (✓)

“Classical” evaluation: forward.
Complexity ⇠ #inputs.

“ @✓ = X @✓ @✓ ”N@✓jrj `(✓)

N

k2Child(j)k@✓k @✓jrk `(✓)@j gk (✓)

Backward evaluation.
Complexity ⇠ #outputs (1 for grad).

Backward Automatic Differentiationp`(✓1 , ✓2 ) = ✓2 e
✓ 1 + ✓ 2 e ✓1
def.

✓1

computing `

✓1
✓2

def.

g 3 ✓3 = e

def.

✓1

g4

def.

g 5 ✓5 = ✓1 + ✓4

g 6 ✓6 =

p

g7

def.

✓4 = ✓2 ✓3

input

✓5
def.

✓7 = ✓4 ✓6

output `function `(✓1 , . . . , ✓M )for r = M + 1, . . . , R✓r = gr (✓Parents(r) )return ✓Rbackward

forwardcomputing r`function r`(✓1 , . . . , ✓M )rR ` = 1

for r = R

1, . . . , 1X
rr ` =@r gs (✓) rs `s2Child(r)return (r1 `, . . . , rM `)Examples of Image Generation(a) WGAN MNIST(a) WGAN MNIST(b) WGAN CelebA(c) WGAN LSUN(a) WGAN MNISTMMDCelebA
GAN MNIST(b)(d)WGAN(

Figure 2: Generated samples from WG
datasets.

digits in Figure 1c are more natural w
both GMMN variants fail to generate m
We observe similar cases in other com
thus results are omitted. On the otherimages with sharp boundary and highproposed
adversarial
learnedkernels
Large
"
Small
"CelebA(d)Inputs
MMD GAN MNIST(e) MMD(f) MMD
LSUN(a) GANWGAN
MNIST(b) GANWGAN
CelebAbetween GMMN and MMD GAN.
p(d) c(x,
MMD
GAN
MNIST
(e)
MMD
GAN
CelebAbedroom (f)FigureGeneratedsamplesfrom
WGAN
and MMDon MNIST,
CelebA,
andLSUN! 2:Needto learnthemetricy)= GAN||d⇠ (x)
d
(y)||
(⇠
GANs)
⇠
If we increase the batch size of GMM
datasets.
Figure 2: ofGeneratedsamplesmodels
from WGAN
MMDGAN
MNIST,! Performance evaluation
generative
anandopen
notiscompetitive
toproblem.
MMDon
GAN
with CBProgressive Growing of GANs for Improved Quality, Stability, and Variation
Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018

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