Entropic continuity bounds for conditional covariances with applications to Schr\" odinger and Sinkhorn bridges
Pierre Del Moral
TL;DR
This work derives an entropic continuity theorem that bounds conditional expectations and covariances in terms of relative entropy between joint bridge distributions, assuming a quadratic transport cost inequality ${\cal T}_2(\rho)$. By combining Wasserstein couplings with ${\cal T}_2(\rho)$ (and LS) inequalities, the authors obtain explicit bounds on ${\cal D}_2(\mu{\cal L},\mu{\cal K})$, as well as on the conditional mean $m$ and conditional covariance $\sigma$ differences, in terms of ${\cal H}(\mu\times{\cal L}||\mu\times{\cal K})$. These entropic bounds are then applied to Schrödinger and Sinkhorn bridges, establishing regularity results and, crucially, yielding simple proofs of exponential decay for the gradients and Hessians of both Schrödinger- and Sinkhorn-bridge potentials. The framework integrates entropic OT (Schrödinger/Sinkhorn) with stability analysis under perturbations, with implications for high-dimensional generative modeling and related ML applications.
Abstract
The article presents new entropic continuity bounds for conditional expectations and conditional covariance matrices. These bounds are expressed in terms of the relative entropy between different coupling distributions. Our approach combines Wasserstein coupling with quadratic transportation cost inequalities. We illustrate the impact of these results in the context of entropic optimal transport problems. The entropic continuity theorem presented in the article allows to estimate the conditional expectations and the conditional covariances of Schr\" odinger and Sinkhorn transitions in terms of the relative entropy between the corresponding bridges. These entropic continuity bounds turns out to be a very useful tool for obtaining remarkably simple proofs of the exponential decays of the gradient and the Hessian of Schrödinger and Sinkhorn bridge potentials.