This section illustrates that (7) holds.

We propose a novel neural algorithm for the fundamental problem of computing the entropic optimal transport (EOT) plan between continuous probability distributions which are accessible by samples. Our algorithm is based on the saddle point reformulation of the dynamic version of EOT which is known as the Schrödinger Bridge problem. In contrast to the prior methods for large-scale EOT, our algorithm is end-to-end and consists of a single learning step, has fast inference procedure, and allows handling small values of the entropy regularization coefficient which is of particular importance in some applied problems. Empirically, we show the performance of the method on several large-scale EOT tasks.

In fact, without such assumptions, this problem is NP-hard [Sim02; MR18].

We study the consistency of surrogate risks for robust binary classification.

Let Cost(π) be the value of weak OT functional for a plan π, i.e., Cost( π) We are going to use our Theorem 3.1. As a result, every plan is optimal.Proof of Proposition 3.3. According to our Theorem 3.2, one only has to ensure that Anyway, this is indifferent for us. It remains to upper bound the first term in (23). Formula (12) for the optimal drift follows from [38, Proposition 4.1] From our Proposition 3.3 it follows that For other ϵ > 0, the analogous equivalence holds true.

This section illustrates that (7) holds.