Stability and upper bounds for statistical estimation of unbalanced transport potentials
Statistics Theory
2022-03-18 v1 Statistics Theory
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.
Cite
@article{arxiv.2203.09143,
title = {Stability and upper bounds for statistical estimation of unbalanced transport potentials},
author = {Adrien Vacher and François-Xavier Vialard},
journal= {arXiv preprint arXiv:2203.09143},
year = {2022}
}