  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 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 a novel methodology which allows utilizing
the recent developments and technical improvements of the former in order to
enrich the latter. From the theoretical perspective, we prove generalization
bounds for our technique. In practice, we validate its applicability in toy 2D
and image domains. To showcase the scalability, we empower our method with a
pre-trained StyleGAN and apply it to high-res AFHQ $512\times 512$ unpaired I2I
translation. For simplicity, we choose simple short- and long-run EBMs as a
backbone of our Energy-guided Entropic OT approach, leaving the application of
more sophisticated EBMs for future research. Our code is available at:
https://github.com/PetrMokrov/Energy-guided-Entropic-OT
