Sliced Inner Product Gromov-Wasserstein Distances

2026-05-08

AI Summary

This work addresses the high computational complexity and poor statistical scalability of Gromov–Wasserstein (GW) distance in aligning high-dimensional heterogeneous data. Focusing on inner-product-based cost functions, it introduces for the first time a sliced GW method that admits a closed-form solution. The proposed approach substantially improves computational efficiency and scalability while preserving key structural properties such as rotational invariance. Theoretical analysis demonstrates favorable geometric characteristics of the sliced formulation, and empirical evaluations on heterogeneous text clustering and language model representation comparison tasks confirm its effectiveness and superiority over existing methods.

Abstract

The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.

Problem

• Gromov-Wasserstein
• scalability
• high-dimensional
• sliced methods
• computational complexity

Innovation

• Sliced Gromov-Wasserstein
• Inner Product Cost
• Rotational Invariance
• Scalable Optimal Transport
• Heterogeneous Data Alignment