Stability and upper bounds for statistical estimation of unbalanced transport potentials

TL;DR

This paper establishes upper bounds on the statistical estimation rates for unbalanced optimal transport maps, leveraging stability properties of the semi-dual formulation, and demonstrates super-parametric convergence rates under certain conditions.

Contribution

It introduces stability-based upper bounds for unbalanced OT potentials and connects local strong convexity to improved statistical rates.

Findings

Derived upper bounds on estimation rates for unbalanced OT maps.

Identified conditions under which super-parametric rates are achievable.

Linked stability properties to convergence speed improvements.

Abstract

In this note, we derive upper-bounds on the statistical estimation rates of unbalanced optimal transport (UOT) maps for the quadratic cost. Our work relies on the stability of the semi-dual formulation of optimal transport (OT) extended to the unbalanced case. Depending on the considered variant of UOT, our stability result interpolates between the OT (balanced) case where the semi-dual is only locally strongly convex with respect the Sobolev semi-norm H1 dot and the case where it is locally strongly convex with respect to the H 1 norm. When the optimal potential belongs to a certain class C with sufficiently low metric-entropy, local strong convexity enables us to recover super-parametric rates, faster than 1 / root n.