Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problem
CoRR(2023)
Abstract
Recently, the Gromov-Wasserstein Optimal Transport (GWOT) problem has
attracted the special attention of the ML community. In this problem, given two
distributions supported on two (possibly different) spaces, one has to find the
most isometric map between them. In the discrete variant of GWOT, the task is
to learn an assignment between given discrete sets of points. In the more
advanced continuous formulation, one aims at recovering a parametric mapping
between unknown continuous distributions based on i.i.d. samples derived from
them. The clear geometrical intuition behind the GWOT makes it a natural choice
for several practical use cases, giving rise to a number of proposed solvers.
Some of them claim to solve the continuous version of the problem. At the same
time, GWOT is notoriously hard, both theoretically and numerically. Moreover,
all existing continuous GWOT solvers still heavily rely on discrete techniques.
Natural questions arise: to what extent existing methods unravel GWOT problem,
what difficulties they encounter, and under which conditions they are
successful. Our benchmark paper is an attempt to answer these questions. We
specifically focus on the continuous GWOT as the most interesting and debatable
setup. We crash-test existing continuous GWOT approaches on different
scenarios, carefully record and analyze the obtained results, and identify
issues. Our findings experimentally testify that the scientific community is
still missing a reliable continuous GWOT solver, which necessitates further
research efforts. As the first step in this direction, we propose a new
continuous GWOT method which does not rely on discrete techniques and partially
solves some of the problems of the competitors. Our code is available at
https://github.com/Ark-130994/GW-Solvers.
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