General Divergence Regularized Optimal Transport: Sample Complexity and Central Limit Theorems

2025/10/02 by Jiaping Yang, Yang, Jiaping, Yunxin Zhang +1
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistical Mechanics and Entropy #Statistics Theory (math.ST) #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.2510.02489

openalex publication_date 2025/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Optimal transport has emerged as a fundamental methodology with applications spanning multiple research areas in recent years. However, the convergence rate of the empirical estimator to its population counterpart suffers from the curse of dimensionality, which prevents its application in high-dimensional spaces. While entropic regularization has been proven to effectively mitigate the curse of dimensionality and achieve a parametric convergence rate under mild conditions, these statistical guarantees have not been extended to general regularizers. Our work bridges this gap by establishing analogous results for a broader family of regularizers. Specifically, under boundedness constraints, we prove a convergence rate of order n^-1/2 with respect to sample size n. Furthermore, we derive several central limit theorems for divergence regularized optimal transport.

Citations

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Related

• Stability and Sample Complexity of Divergence Regularized Optimal Transport
• Sparse Regularized Optimal Transport without Curse of Dimensionality
• Convergence Rates for Regularized Optimal Transport via Quantization
• Sharp Asymptotics for Regularized Optimal Transport
• Interpolating between Optimal Transport and KL regularized Optimal Transport using Rényi Divergences