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 Schrödinger Gromov X <latexit 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<latexit sha1_base64="6tOGw6FSQVfHa06oZqSaTlaRZAo=">AABFdnictVxtc9u4EUaub9f0Ldd+7EyHPZ/bXCd17TSdduamM5fYjuOLkziR7CQXJRlRohUltKiIkvKi81/o1/bf9Hf0H7Sf+rUfuy8AAUogF3TTcGyDIJ7dxRJY7C7AxON0mE83N/9x4aNvffs73/3ex9+/+IMf/ujHP7n0yU+P82w26SVHvSzNJo/ibp6kw1FyNB1O0+TReJJ0T+M0eRi/2sbnD+fJJB9mo/b03Th5etodjIYnw153ilWdx++2n19a29zYpH/RamFLF9aU/neYffLphuqovspUT83UqUrUSE2hnKquyuF6orbUphpD3VO1gLoJlIb0PFFn6iJgZ9AqgRZdqH0Fvwdw90TXjuAeaeaE7gGXFH4mgIzUOmAyaDeBMnKL6PmMKGNtFe0F0UTZ3sHfWNM6hdqpegG1Es60DMVhX6bqRP2J+jCEPo2pBnvX01RmpBWUPHJ6NQUKY6jDch+eT6DcI6TRc0SYnPqOuu3S839SS6zF+55uO1P/IinX4YpUS/c+Kyh01ZzoR/Q2Z/CM5UmB8wAoJLqPWHpDuj6l3o+g/QLq78J1RiWjkxiuBdWe1SK34fIht0XkHlw+5J6IPIDLhzwQkYdw+ZCHGonYCencj2/B5cO3RM734fIh74vIB3D5kA9E5DFcPuSxiPwaLh/yaxF5Ey4f8qaIvA2XD3lbRLbh8iHbIvIILh/ySETuwuVD7mpk9UydwJURnaEwK69DucwDLUUKNddF+W6QdfRhbwTM6V4FVp7VO/DXj90J0GlSgd0NGHcnFVh55O2BjfRjZVt0i1YTH/aWiN2HEeDH7ovYr9TLCuxXATPtVQVWnmsH0M6Pla3vHbjzY++I2LtQ8mPlNeoe1Pix9wJWjHEF9lDE3levK7AhVn9SgZXtfgvsih8rr1NtaO/HhljTWQVWtqfH4MH4sfJq9RBq/diHIvaReluBfSRiH4N192MfB6yw7yuwZo29SCvIgPyRBGZsHbVuMSuxNAZqXYF/WqwtKfnGMdRLmEGBGRDmVETsFYi9QMRBgTgIlisv7GhO/q7MpVUgWoGIuFibsDQV2/eL9lhKAxA7BWJnCVHnkeK7Nn2Zk3dhaiTktFi5sBTSp6yw31hK9Hiot7wGca+E4LH9gkb+FYqWMIJCTdVRe1Gs8YyM6L4O8YaiN9NLw0PGTQur4KLeiqjYg4pF1DsP6p2ImnlQMxE196DmIsrOfBfXCRgBVv/4LhZ0xyOAfeTqKwKv4DqsOrdgjkYwfg7BC3xANffgb4tib+mqkwyjeVwnMcvxtGSJJ1BaqDWot1HhDsXXKc2wBCTjlvd0jI93mNtY6DnHVvisWMmjImMSTmdI8gwKOugtRjSfmtG5TTVn5N1xqRn+VjHvTakZfpc0fkZePJea4ada+uk5ZG9rbPsc2BbMprHWvi03pcH5F6Zhyhdp1UWLi2/1VI8ZpPe2If19/Wb2z/FetqnE+rHlZjRyp395qX9NaFg9546em1FB74m9XlOKGvdkpONeW24qQ0ar6EjLYe+avhls09dvxpSb0TgEj2ubYu6FU246esdFb2y5GY1jxXnPM/LkTbkZjQHdsz5suRkNzLZ0dZxvy00tO2qAY2dbbmrVR5QFxhwQj3musV7RhPykmaY2JP+gPlvj+vyr6xjmbJ4VMUI9JevbVtOJi7WsXiLjLyRg1aYN5UD/Yub4YGUaC3VVjK9YhmlpfV+lY9d41PwBaDGC2c97AFLOPAUJTU4CrXcKFLfEqKvcM4O7KuJwlJwsoTq6dip6i5YvZ43Kdc+pVorLbG+tHjtkr3Mae2PyCQ9Is5IeDirfcBVFSUMHJQ3J9Jro7r2er2Xtb4q48RJiXIy0Hu0I8U5afZzq03rL0fG63uWZwsV7Pnb8Yrb5RFsbjHkyskUoSx1Pt53JI7l1uK5eUTbHzc8ieqNor+ZkNYa0I5WLUajJFrM3vqB7S/uI9uSQB9PowXuMNJWx4l0zzKJjPj0ii+raW4k36stk6Lick9U19rgePXDQAw+6eYyzDSvGXSi1IWY4grt2QJRzsdBVRhqfqN8Wu6MZvcH6iD4tWUhDg+1NUrKQdVH2ixKVN4DG0cBRejiNZToG31mhJEf9Pnls7Fq2/Ou0c2v2t7s0xqtHc3Umpk9crxLXiGYN7+ry3TIHlmDhfXKV/Nf6XiK/JhzRhkpcnzmcWS8j2vFPKIIdk2ec0myTZke5tZufWn5iOB0qs3eOu9kZWciI7F8E61NGYzKiH/fsgNlBZ4uQko0MsTvDwrvx+TpDcYxZP26o+FSDHW8J2bIZ8Td03dmV01jkiIHXgbOlsW10ckC+YEJcJ9q627ldv/og0p6TcEcJU7Rj5TLx/5x+mx8zTtZWRgRqGN9Arm2d731kFLOgjrq0ytfbINPWlfKzQoZnWmq7/lmZPitJtkMRF8qDq3UfOPfonnnhKJmQ3PlKG15H67K5SHm8pEfs7QlF8Wz3B3oFRrmv0Cq5RnOuQ6NkAKNgWkQRpq2URV7mW8+rTD2Mdv5/oW51XdYaUoyUzeCyhqT8fkLRmitlCqOax+8rmk1+rU+WWtXzGdFYPHXm8jdQ+0v4beQ292F04pJVuEFjgCnYO6sRrolWWoTxulHiZUamoWXvLT87Jk0rt+Y88TVbNxtjzxtTOaRR81ZnLUz5PDReOjReBuqwTXuNVoum3lii52Js0da7laH8mnBrN6A8EynLHplBDQOkdGOpMKp9kaoc4xvUe5HWpkirC7PV3Q1w53wI0j/Xl2f3N8XqHqmb5Nv0yAPj+KVPs3RIPpeprY/UmAJyvqbtqzv7O1SD3GOyoEiZz3HijOFdpx5dZ4Wkv9IrW0Z23loEc27pjW5jbGyHyr9fQZ7SnMhpXhrENWqRaPldOaIli7Th+BwRZf675FOx31EfM7ut7TuJSv6EjTd5VlleHCmMSP9S5m1/JXrdd+LXiGLCmfauY6DV/A0jBcaYTILfs8zpDeEqxzsJ7NHGZD9X7RTv4o0ciTZI6oX6c4CN4ajXjnV3bJkem779Blqi1u1b97WQ+aXBHCV+59nR69Kqdqp91MXS/flodfUqV76v08Nsia/Vx4zauJGFjfLKmI76IpgLS9SMC2NCuDTrRRP5m0neRGbenQqlbFobyuVMA9uYFxQvSedAEeHz7i57vbnPhX7EK/RiwrrUuEaihNm4TOcHXEuLWaloKUJy66U1KXXWo6r1wvJwVw1rx9lSJmQFUyXlbri124dOKVqRszFMoaf4ZG9VnOjS/AIu/B0pX5RoOIbkEFvg515X22r3A5yKeK3LnNmMqAZtQn8pBu/qfpZb1OvotUPdpR/CIZzHEHQtST+kFbWp7ExZltylHk7/DVmDiUpE6W3L5n1wucg9WeXUpD9DsnByb4bKfJPTtC+GQ0hPylzC+fD+htSLE2W+bWrWB0Nd7kGZQxMe5jxD2Du3rZvzcjnV62uVSygPXgfMzovB4Q5gdcxi24VYqInzRj48B7QOJzXUzWrxv/bD8LGcmvMK5ZbTN2cvA946t0t0Zhb94uZzxnILGc3VHMN5ZkXvrNfk58f+X9ToTWVObz48ffRL7RgwvBaK86GydIx3R5GVN5QK7g/4ZMjUv9XfL8hfJbwuaFTJ0YSS2a+opmZayNTMl5e+3plnITJZOlUylanZeKJFJ2O31b66CT/bhQfY9JQof1PJfxHr/462D7UnZD1MNp0zCB2qSygLYnfT+nRvz9FWSYxnevmMbxtqcE/8gGrxvO9dao9nftulvlV/ScJz/Y7KVL8UmSzv8tl5FUMPyjtwnAsy3/tGdKaes1l8Au00YI+Rz1FxpGS+fl4Qok9x4bKkC0KY0VJHOfZSjulMUlJBOy71rUcjfKx3+nHfAc/nd4vsUqR+R3VdvTrgSi1JdeiR6gllBmLS/yZEaH9QV+DvFV32S3q4ImlO76As0VvnWf1JsDPvuLBfM65THsxk6ua6XUZRvd09rM/E7lRy4RPv9fhBDX7gSNmit/WK4u6Jqs8dzmpozrRM7n7uSJm8J+sBo9luMT7q4+d5Da95QP9vV6JvO5LugSwxZdsj2s+bEL1U62aXpOdzlfV521s10pqvNpmmPVlpx4E5I1nHAb8q2xZmP395Vr8y4BdmfjruXH8csO+CPZEk4pOZUvYoCZCIz4hK51eGXkqyxRgHnM/oBvRW7mtITyUqM1GSWcC30fMAWeYBdE4EaU5ECgNREm2xnl9a21r+30dWC8dXN7Y2N7buX1v78ob+n0k+Vj9Xn6rLsBr/UX0JM/JQHVGu8S/qr+pvO//Z/cXu+u6vuelHFzTmZ6r0b3fzv4JTXQQ=</latexit> convergence Sinkhorn and