February 2024 On the difference between entropic cost and the optimal transport cost
Soumik Pal
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Ann. Appl. Probab. 34(1B): 1003-1028 (February 2024). DOI: 10.1214/23-AAP1983
Abstract

Consider the Monge–Kantorovich problem of transporting densities ρ0 to ρ1 on Rd with a strictly convex cost function. A popular regularization of the problem is the one-parameter family called the entropic cost problem. The entropic cost Kh, h>0, is significantly faster to compute and hKh is known to converge to the optimal transport cost as h goes to zero. We are interested in the rate of convergence. We show that the difference between Kh and 1/h times the optimal cost of transport has a pointwise limit when transporting a compactly supported density to another that satisfies a few other technical restrictions. This limit is the relative entropy of ρ1 with respect to a weighted Riemannian volume measure on Rd that measures the local sensitivity of the transport map. For the quadratic Wasserstein transport, this relative entropy is exactly one half of the difference of entropies of ρ1 and ρ0. More surprisingly, we demonstrate that this difference of two entropies (plus the cost) is also the limit for the Dirichlet transport introduced by Pal and Wong (Probab. Theory Related Fields 178 (2020) 613–654) in the context of stochastic portfolio theory. The latter can be thought of as a multiplicative analog of the Wasserstein transport and corresponds to a nonlocal operator. The proofs are based on Gaussian approximations to Schrödinger bridges as h approaches zero.

Copyright © 2024 Institute of Mathematical Statistics
Soumik Pal "On the difference between entropic cost and the optimal transport cost," The Annals of Applied Probability 34(1B), 1003-1028, (February 2024). https://doi.org/10.1214/23-AAP1983
Received: 1 August 2021; Published: February 2024
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Vol.34 • No. 1B • February 2024
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