On the difference between entropic cost and the optimal transport cost

Abstract

Consider the Monge–Kantorovich problem of transporting densities ${\mathit{\rho }}_0$ to ${\mathit{\rho }}_1$ on ${\mathbb{R}}^{\mathit{d}}$ 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 ${\mathit{K}}_{\mathit{h}}$, $\mathit{h}>0$, is significantly faster to compute and $\mathit{h}{\mathit{K}}_{\mathit{h}}$ 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 ${\mathit{K}}_{\mathit{h}}$ and $1/\mathit{h}$ 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 ${\mathit{\rho }}_1$ with respect to a weighted Riemannian volume measure on ${\mathbb{R}}^{\mathit{d}}$ 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 ${\mathit{\rho }}_1$ and ${\mathit{\rho }}_0$. 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.