Sum-of-Squares Hierarchy for the Gromov Wasserstein Problem

TL;DR

This paper introduces semidefinite relaxations based on the Sum-of-Squares hierarchy for the Gromov-Wasserstein problem, enabling tractable computation of approximate solutions and distances between metric measure spaces.

Contribution

It develops simplified SOS hierarchies with convergence guarantees for solving the GW problem and defining new distances that satisfy the triangle inequality.

Findings

Proposes tractable SOS relaxations for GW problem.

Proves convergence rates of the hierarchies.

Defines new distances satisfying metric properties.

Abstract

The Gromov-Wasserstein (GW) problem is a variant of the classical optimal transport problem that allows one to compute meaningful transportation plans between incomparable spaces. At an intuitive level, it seeks plans that minimize the discrepancy between metric evaluations of pairs of points. The GW problem is typically cast as an instance of a non-convex quadratic program that is, unfortunately, intractable to solve. In this paper, we describe tractable semidefinite relaxations of the GW problem based on the Sum-of-Squares (SOS) hierarchy. We describe how the Putinar-type and the Schm\"udgen-type moment hierarchies can be simplified using marginal constraints, and we prove convergence rates for these hierarchies towards computing global optimal solutions to the GW problem. The proposed SOS hierarchies naturally induce a distance measure analogous to the distortion metrics, and we show that these are genuine distances in that they satisfy the triangle inequality. In particular, the proposed SOS hierarchies provide computationally tractable proxies of the GW distance and the associated distortion distances (over metric measure spaces) that are otherwise intractable to compute.