Optimal Transport in the Face of Noisy Data

Optimal transport distances are popular and theoretically well understood in the context of data-driven prediction. A flurry of recent work has popularized these distances for data-driven decision-making as well although their merits in this context are far less well understood. This in contrast to the more classical entropic distances which are known to enjoy optimal statistical properties. This begs the question when, if ever, optimal transport distances enjoy similar statistical guarantees. Optimal transport methods are shown here to enjoy optimal statistical guarantees for decision problems faced with noisy data.