Robust Alignment via Partial Gromov-Wasserstein Distances

2025-06-26

AI Summary

Gromov–Wasserstein (GW) alignment is highly sensitive to outliers in heterogeneous data, leading to severe structural distortion. To address this, we propose a robust alignment framework based on the Partial GW (PGW) distance: by truncating distributional mass, PGW explicitly suppresses outlier influence; we establish PGW as the first robust surrogate of classical GW and prove its minimax optimality. We further reveal that PGW induces an approximate pseudometric structure, endowing it with clear statistical interpretability. Theoretically, our method achieves (near-)minimax optimal estimation both in the population and finite-sample regimes under arbitrary contamination. Empirically, it significantly improves alignment accuracy across diverse heterogeneous structural data. This work provides a principled, theoretically grounded paradigm for robust geometric learning.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Intelligent Robots: Learning & Optimization for ROB
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Robust alignment of heterogeneous datasets with outliers
• Estimating GW alignment cost between uncontaminated measures
• Partial GW distance as robust surrogate for classical distance

Innovation

• Partial GW distance for robust alignment
• Trims mass to handle outliers effectively
• Minimax optimal estimator for contamination