“…Some of the following properties were already known for the softmax operator (for example in [39]) and they are used in order to get a posteriori regularity of the potentials but, up to our knowledge, were never used to get a priori results. Another very cleverly used properties of the (c, ε)-transform are in [32] in order to obtain a new proof of the Caffarelli's contraction theorem [12].…”
“…[41,45]), optimal transport theory (e.g. [5,16,18,20,21,32,44,51,54,55,58]), data sciences (e.g. [34,39,40,56,57,62,63] see also the book [28] and references therein).…”
This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.
“…Some of the following properties were already known for the softmax operator (for example in [39]) and they are used in order to get a posteriori regularity of the potentials but, up to our knowledge, were never used to get a priori results. Another very cleverly used properties of the (c, ε)-transform are in [32] in order to obtain a new proof of the Caffarelli's contraction theorem [12].…”
“…[41,45]), optimal transport theory (e.g. [5,16,18,20,21,32,44,51,54,55,58]), data sciences (e.g. [34,39,40,56,57,62,63] see also the book [28] and references therein).…”
This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.
“…This gives the backward decomposition: given two probability measures, there is a transport plan between them given by the gradient map of a convex contraction, followed by a martingale coupling. This establishes a link with the celebrated Caffarelli's contraction theorem [15] (see also [30] and [18]) : if ν is a log-concave perturbation of the Gaussian measure µ, then the optimal transport map from µ to ν is given by the gradient of a convex function which is a contraction. In the language of item (2), the optimal map is a contraction when the projection μ to the backward cone is equal to ν.…”
We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their characterizations require study on a case-by-case basis. Particular attention is given to convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. Our results extend to the forward convex order case the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to the early results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.
“…The existence of a connection between majorization and Caffarellli's contraction theorem seems to go back to Hargé [14] (see [12,11] for more recent results in this direction). Here we may put together our transport approach of the majorization in Lemma 2.4 and the latter theorem to get the following natural statement.…”
We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all orders among all strongly log-concave densities.
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