ENOT: Expectile Regularization for Fast and Accurate Training of Neural
  Optimal Transport
Nazar Buzun Maksim Bobrin Dmitry V. Dylov
We present a new extension for Neural Optimal Transport (NOT) training
procedure, capable of accurately and efficiently estimating optimal
transportation plan via specific regularisation on conjugate potentials. The
main bottleneck of existing NOT solvers is associated with the procedure of
finding a near-exact approximation of the conjugate operator (i.e., the
c-transform), which is done either by optimizing over maximin objectives or by
the computationally-intensive fine-tuning of the initial approximated
prediction. We resolve both issues by proposing a new, theoretically justified
loss in the form of expectile regularization that enforces binding conditions
on the learning dual potentials. Such a regularization provides the upper bound
estimation over the distribution of possible conjugate potentials and makes the
learning stable, eliminating the need for additional extensive finetuning. We
formally justify the efficiency of our method, called Expectile-Regularised
Neural Optimal Transport (ENOT). ENOT outperforms previous state-of-the-art
approaches on the Wasserstein-2 benchmark tasks by a large margin (up to a
3-fold improvement in quality and up to a 10-fold improvement in runtime).