We present a new approach for Neural Optimal Transport (NOT) training procedure, capable of accurately and efficiently estimating optimal transportation plan via specific regularization on dual Kantorovich 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 non-convex max-min 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 which enforces binding conditions on the learning process of the dual potentials. Such a regularization provides the upper bound estimation over the distribution of possible conjugate potentials and makes the learning stable, completely eliminating the need for additional extensive fine-tuning. Proposed method, called Expectile-Regularized Neural Optimal Transport (ENOT), outperforms previous state-of-the-art approaches in the established 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). Moreover, we showcase performance of ENOT for various cost functions in different tasks, such as image generation, demonstrating generalizability and robustness of the proposed algorithm.

This paper derives a new strong Gaussian approximation bound for the sum of independent random vectors. The approach relies on the optimal transport theory and yields explicit dependence on the dimension size p and the sample size n. This dependence establishes a new fundamental limit for all practical applications of statistical learning theory. Particularly, based on this bound, we prove approximation in distribution for the maximum norm in a high-dimensional setting (p > n).

In many sequential tasks, a model needs to remember relevant events from the distant past to make correct predictions. Unfortunately, a straightforward application of gradient based training requires intermediate computations to be stored for every element of a sequence. This requires to store prohibitively large intermediate data if a sequence consists of thousands or even millions elements, and as a result, makes learning of very long-term dependencies infeasible. However, the majority of sequence elements can usually be predicted by taking into account only temporally local information. On the other hand, predictions affected by long-term dependencies are sparse and characterized by high uncertainty given only local information. We propose \texttt{MemUP}, a new training method that allows to learn long-term dependencies without backpropagating gradients through the whole sequence at a time. This method can potentially be applied to any recurrent architecture. LSTM network trained with \texttt{MemUP} performs better or comparable to baselines while requiring to store less intermediate data.

We propose a new unsupervised and non-parametric method to detect change points in electrocardiography. The detection relies on optimal transport theory combined with topological analysis and the bootstrap procedure. The algorithm is designed to detect changes in virtually any harmonic or a partially harmonic signal and is verified on ECG data streams. We successfully find abnormal or irregular cardiac cycles in the waveforms for the six of the most frequent types of clinical arrhythmias using a single algorithm. Our unsupervised approach reaches the level of performance of the supervised state-of-the-art techniques. We provide conceptual justification for the efficiency of the method.