Contraction of the proximal map and generalized convexity of the Moreau-Yosida regularization in the 2-Wasserstein metric

TY - JOUR

T1 - Contraction of the proximal map and generalized convexity of the Moreau-Yosida regularization in the 2-Wasserstein metric

AU - Carlen, Eric A.

AU - Craig, Katy

N1 - Funding Information: Carlen's work was partially supported by U.S. National Science Foundation grant DMS 0901632. Craig's work was partially supported by a Presidential Fellowship at Rutgers University. Publisher Copyright: © 2013 Mathematical Sciences Publishers.

PY - 2013

Y1 - 2013

N2 - We investigate the Moreau-Yosida regularization and the associated proximal map in the context of discrete gradient flow for the 2-Wasserstein metric. Our main results are a stepwise contraction property for the proximal map and an "above the tangent line" inequality for the regularization. Using the latter, we prove a Talagrand inequality and an HWI inequality for the regularization, under appropriate hypotheses. In the final section, the results are applied to study the discrete gradient flow for Rényi entropies. As Otto showed, the gradient flow for these entropies in the 2-Wasserstein metric is a porous medium flow or a fast diffusion flow, depending on the exponent of the entropy. We show that a striking number of the remarkable features of the porous medium and fast diffusion flows are present in the discrete gradient flow and do not simply emerge in the limit as the time-step goes to zero.

AB - We investigate the Moreau-Yosida regularization and the associated proximal map in the context of discrete gradient flow for the 2-Wasserstein metric. Our main results are a stepwise contraction property for the proximal map and an "above the tangent line" inequality for the regularization. Using the latter, we prove a Talagrand inequality and an HWI inequality for the regularization, under appropriate hypotheses. In the final section, the results are applied to study the discrete gradient flow for Rényi entropies. As Otto showed, the gradient flow for these entropies in the 2-Wasserstein metric is a porous medium flow or a fast diffusion flow, depending on the exponent of the entropy. We show that a striking number of the remarkable features of the porous medium and fast diffusion flows are present in the discrete gradient flow and do not simply emerge in the limit as the time-step goes to zero.

KW - Gradient flow

KW - Moreau-Yosida regularization

KW - Wasserstein metric

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U2 - https://doi.org/10.2140/memocs.2013.1.33

DO - https://doi.org/10.2140/memocs.2013.1.33

M3 - Article

SN - 2326-7186

VL - 1

SP - 33

EP - 65

JO - Mathematics and Mechanics of Complex Systems

JF - Mathematics and Mechanics of Complex Systems

IS - 1

ER -