Sample complexity of unbalanced entropic OT

2026-06-23

AI Summary

This work addresses the rigidity of classical optimal transport in scenarios where mass conservation does not hold and its poor sample complexity in high dimensions. The authors investigate the finite-sample theory of entropy-regularized unbalanced optimal transport, introducing a translation-invariant dual formulation and analyzing its geometric structure. They establish, for the first time, high-probability convergence bounds for empirical optimal couplings at the coupling level. Their analysis demonstrates that entropy regularization not only substantially mitigates the curse of dimensionality and reduces the required sample size but also enhances estimation stability, all while preserving compatibility with efficient solvers such as Sinkhorn algorithms.

Abstract

Optimal transport (OT) has become a central language for comparing probability measures, but exact balanced OT is often both too rigid for data with missing, created, or destroyed mass and subject to unfavorable high-dimensional sample complexity. Entropic regularization and unbalanced relaxations address these limitations in complementary ways. Entropy smooths the geometry, improves statistical behavior, and enables fast Sinkhorn-type algorithms, while unbalanced marginal penalties replace hard conservation constraints by divergence terms adapted to noisy empirical data. This paper studies the sample complexity of entropic unbalanced OT at the level of the optimal coupling, rather than only the scalar transport value. We develop a translation-invariant dual formulation, prove compactness and strong convexity properties for the intrinsic dual variables, and convert these geometric estimates into high-probability finite-sample bounds for empirical couplings. The results clarify why regularization is a practical necessity in machine learning applications: it softens the curse of dimensionality, reduces the number of samples needed for stable transport estimation, and keeps the resulting estimators compatible with scalable Sinkhorn-type solvers.

Problem

• sample complexity
• unbalanced optimal transport
• entropic regularization
• optimal coupling
• high-dimensional statistics

Innovation

• entropic regularization
• unbalanced optimal transport
• sample complexity
• dual formulation
• Sinkhorn algorithm