Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime

2026-09-22

AI Summary

研究半离散情形下熵最优传输的统计速率，通过强凹性分析和特化边界方法，实现参数收敛率及快速下游量估计。

Abstract

We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate $n^{-1}$, matching the fully compact case and improving over the less favorable $n^{-1/2}$ rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove $\sqrt{n}$-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a $\sqrt{\log n}$ factor. Simulations support the theory.

Problem

• Entropic Optimal Transport
• Semi-Discrete Regime
• Statistical Rates
• Dual Potentials
• Barycentric Projection

Innovation

• parametric convergence rates
• semi-discrete regime
• barycentric projection
• Sinkhorn-EM