Sum-of-Squares

A focus on Sinkhorn algorithm, an improvement in the unbalanced setting, and the curse of dimension.

After presenting some background on optimal transport, we present entropic regularization, its link with the Schr"odinger problem and the Sinkhorn algorithm. We present unbalanced optimal transport and the corresponding Sinkhorn algorithm. Then, we show how to improve on the vanilla Sinkhorn algorithm in this particular case. Then, we switch to a different problem which is the statistical estimation of optimal transport. Under smoothness assumptions on the transport maps we achieve a parametric rate of estimation of the distance using a sum-of-squares in Sobolev spaces.