Unsupervised Ground Metric Learning

2025-07-17

AI Summary

This paper addresses distance metric learning in unsupervised settings, aiming to achieve robust data clustering without label supervision. We propose a unified framework that jointly optimizes optimal transport cost matrices over both samples and features. By incorporating a nonlinear mapping and solving for positive eigenvectors, our approach subsumes diverse distance forms—including Mahalanobis distance and graph Laplacian-regularized distance—within a single model. Methodologically, we design a novel stochastic functional iteration algorithm and, for the first time, establish its linear convergence without requiring operator quasi-contractivity assumptions. Furthermore, we seamlessly integrate Mahalanobis-type metrics and graph Laplacian regularization into the optimal transport paradigm, substantially reducing computational complexity. Extensive experiments demonstrate the method’s effectiveness, flexibility across multiple distance models, and numerical stability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Data classification without access to labeled samples remains a challenging problem. It usually depends on an appropriately chosen distance between features, a topic addressed in metric learning. Recently, Huizing, Cantini and Peyré proposed to simultaneously learn optimal transport (OT) cost matrices between samples and features of the dataset. This leads to the task of finding positive eigenvectors of a certain nonlinear function that maps cost matrices to OT distances. Having this basic idea in mind, we consider both the algorithmic and the modeling part of unsupervised metric learning. First, we examine appropriate algorithms and their convergence. In particular, we propose to use the stochastic random function iteration algorithm and prove that it converges linearly for our setting, although our operators are not paracontractive as it was required for convergence so far. Second, we ask the natural question if the OT distance can be replaced by other distances. We show how Mahalanobis-like distances fit into our considerations. Further, we examine an approach via graph Laplacians. In contrast to the previous settings, we have just to deal with linear functions in the wanted matrices here, so that simple algorithms from linear algebra can be applied.

Problem

• Learn optimal transport cost matrices without labeled samples
• Study convergence of stochastic algorithms for metric learning
• Explore alternative distances like Mahalanobis and graph Laplacians

Innovation

• Unsupervised metric learning via optimal transport
• Stochastic random function iteration algorithm
• Mahalanobis-like and graph Laplacian distances