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

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.

Statistics Theory (math.ST); Machine Learning (cs.LG)

arXiv:2609.26647 [math.ST]

(or arXiv:2609.26647v1 [math.ST] for this version)

https://doi.org/10.48550/arXiv.2609.26647 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.