Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime
Paper proves parametric rates for semi-discrete entropic optimal transport, including barycentric projection and Sinkhorn-EM.
The paper studies statistical rates for entropic optimal transport when one measure has finite support and the other is subGaussian. Empirical dual potentials converge to population counterparts at parametric rates, with no dimension dependence in the leading term. The empirical barycentric projection achieves squared-error rate n^{-1}, improving on the n^{-1/2} rate known for fully subGaussian settings. For a balanced two-component Gaussian mixture, Sinkhorn-EM iterates are sqrt(n)-consistent for any fixed iteration count, up to a sqrt(log n) factor, and simulations support the theory.
- Parametric dual-potential rates have no leading dimension dependence.
- Barycentric projection reaches squared error n to the minus one.
- Sinkhorn-EM is sqrt(n)-consistent up to a sqrt(log n) factor.
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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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.26647