The NP-hardness of the Gromov-Wasserstein distance

TL;DR

The details on the non-convex nature of the GW optimization problem that imply NP-hardness of the GW distance between finite spaces for any instance of an input data are provided.

Abstract

This note addresses the property frequently mentioned in the literature that the Gromov-Wasserstein (GW) distance is NP-hard. We provide the details on the non-convex nature of the GW optimization problem that imply NP-hardness of the GW distance between finite spaces for any instance of an input data. We further illustrate the non-convexity of the problem with several explicit examples.

Figures (1)

• Figure 1: Two additional examples illustrating the presence of negative eigenvalues in $\Gamma_p$, as described in Example \ref{['ex:additional']}. A. Top panel: a pair of spaces from the family constructed by Memoli (Example 5.2). Bottom panel: Fixing the first space with $m=2$ points and varying the number of points $n$ of the second space results in the $\Gamma_p$, $p=1$, having increasing number of negative eigenvalues. B. Top panel: a pair of spaces constructed in Kravtsova using the model from Xiao ($m=n=500$ points in each space is shown). Bottom panel: Fixing the first space with $m=50$ and varying number of points in the second space results in $\Gamma_p$, $p=1$, having increasing number of negative eigenvalues.

Theorems & Definitions (5)

• Definition 1: GW distance (finite spaces case), Definition 5.7 of Memoli
• Example 3
• Theorem 4: GW distance between finite spaces is NP-hard
• proof
• Example 6: Two additional illustrations for negative eigenvalues of $\Gamma_p$, Figure \ref{['fig:GW']}