Sum-of-Squares Hierarchy for the Gromov Wasserstein Problem

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 Schmüdgen-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.

Figures (5)

• Figure 1: Selected poses of cats from the TOSCA dataset bronstein2008numerical.
• Figure 2: Selected poses of dogs from the TOSCA dataset bronstein2008numerical.
• Figure 3: Results of the dog-shapes experiments
• Figure 4: Selected poses of a human from the TOSCA dataset bronstein2008numerical.
• Figure 5: Results of the human-shapes experiments

Theorems & Definitions (14)

• Theorem 3.1
• proof : Proof of Theorem \ref{['connection between 2 hierarchies']}
• Proposition 3.2
• proof : Proof of Proposition \ref{['thm:strongduality']}
• Theorem 4.1
• Theorem 4.2
• Lemma 4.3: bergthaller1992distance,Theorem 0.1
• proof : Proof of Theorem \ref{['thm: convergence rate']}
• Proposition 5.1
• proof