Estimation of entropy-regularized optimal transport maps between
  non-compactly supported measures
Matthew Werenski James M. Murphy Shuchin Aeron
This paper addresses the problem of estimating entropy-regularized optimal
transport (EOT) maps with squared-Euclidean cost between source and target
measures that are subGaussian. In the case that the target measure is compactly
supported or strongly log-concave, we show that for a recently proposed
in-sample estimator, the expected squared $L^2$-error decays at least as fast
as $O(n^{-1/3})$ where $n$ is the sample size. For the general subGaussian case
we show that the expected $L^1$-error decays at least as fast as $O(n^{-1/6})$,
and in both cases we have polynomial dependence on the regularization
parameter. While these results are suboptimal compared to known results in the
case of compactness of both the source and target measures (squared $L^2$-error
converging at a rate $O(n^{-1})$) and for when the source is subGaussian while
the target is compactly supported (squared $L^2$-error converging at a rate
$O(n^{-1/2})$), their importance lie in eliminating the compact support
requirements. The proof technique makes use of a bias-variance decomposition
where the variance is controlled using standard concentration of measure
results and the bias is handled by T1-transport inequalities along with sample
complexity results in estimation of EOT cost under subGaussian assumptions. Our
experimental results point to a looseness in controlling the variance terms and
we conclude by posing several open problems.