Energy-guided Entropic Neural Optimal Transport

Energy-Based Models (EBMs) are known in the Machine Learning community for decades. Since the seminal works devoted to EBMs dating back to the noughties there have been appearing a lot of efficient methods which solve the generative modelling problem by means of energy potentials (unnormalized likelihood functions). In contrast, the realm of Optimal Transport (OT) and, in particular, neural OT solvers is much less explored and limited by few recent works (excluding WGAN based approaches which utilize OT as a loss function and do not model OT maps themselves). In our work, we bridge the gap between EBMs and Entropy-regularized OT. We present the novel methodology which allows utilizing the recent developments and technical improvements of the former in order to enrich the latter. From the theoretical perspectives, we prove generalization bounds for our approach. In practice, we validate its applicability on toy 2D scenarios as well as standard unpaired image-to-image translation problems. For simplicity, we choose simple long-run EBMs as a backbone of our Energy-guided Entropic OT method, leaving the application of more sophisticated EBMs for future research.