Robust Alignment via Partial Gromov-Wasserstein Distances

TL;DR

This paper introduces a robust partial Gromov-Wasserstein estimator that effectively aligns datasets despite contamination and outliers, providing theoretical guarantees and operational interpretation.

Contribution

It proposes a minimax optimal partial GW estimator that trims outliers, with new structural results on the partial GW distance's pseudo-metric properties.

Findings

Estimator is minimax optimal in the population setting.

Near-optimal in finite-sample regimes with small optimality gap.

Provides an operational interpretation of partial GW as a robust alignment measure.

Abstract

The Gromov-Wasserstein (GW) problem provides a powerful framework for aligning heterogeneous datasets by matching their internal structures in a way that minimizes distortion. However, GW alignment is sensitive to data contamination by outliers, which can greatly distort the resulting matching scheme. To address this issue, we study robust GW alignment, where upon observing contaminated versions of the clean data distributions, our goal is to accurately estimate the GW alignment cost between the original (uncontaminated) measures. We propose an estimator based on the partial GW distance, which trims out a fraction of the mass from each distribution before optimally aligning the rest. The estimator is shown to be minimax optimal in the population setting and is near-optimal in the finite-sample regime, where the optimality gap originates only from the suboptimality of the plug-in estimator in the empirical estimation setting (i.e., without contamination). Towards the analysis, we derive new structural results pertaining to the approximate pseudo-metric structure of the partial GW distance. Overall, our results endow the partial GW distance with an operational meaning by posing it as a robust surrogate of the classical distance when the observed data may be contaminated.