2023/06/12 by Hugo Malamut, Malamut, Hugo, Maxime Sylvestre +1 · 2 citations
Mathematics · #49K40 #49Q22 #94A17 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2306.06940
openalex publication_date 2023/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We study the convergence of the transport plans γε towards γ0 as well as the cost of the entropy-regularized optimal transport (c,γε) towards (c,γ0) as the regularization parameter ε vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance W2(γε,γ0) is asymptotically greater than C√ε and the suboptimality (c,γε)-(c,γ0) is of order ε. In the quadratic cost case the compactness assumption is relaxed into a moment of order 2+δ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance W2(γε,γ0) converges to 0 at rate √ε. Finally, if in addition the marginals have finite Fisher information, we prove (c,γε)-(c,γ0) ∼ dε/2 and we provide a companion expansion of H(γε). These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.