Geometric medians on product manifolds

2025-05-24

AI Summary

This work addresses the theoretical absence and computational intractability of geometric medians for heterogeneous geometric variables jointly modeled on product manifolds. We establish, for the first time, a rigorous framework for existence, uniqueness, and local Lipschitz stability of geometric medians on product manifolds, proving global uniqueness and optimal robustness on Hadamard product manifolds. We propose two novel algorithms: a Riemannian subgradient method and a product-aware Weiszfeld iteration—both guaranteed to converge globally with local linear convergence rates, thereby resolving a long-standing gap in median analysis under positive-curvature factors. Our analysis integrates Bures–Wasserstein geometry with curvature and injectivity radius constraints. Experiments on canonical settings—including Gaussian parameter manifolds—demonstrate that the proposed median estimator significantly outperforms the Fréchet mean, exhibiting superior robustness against outliers and data corruption.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

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
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Product manifolds arise when heterogeneous geometric variables are recorded jointly. While the Fr'{e}chet mean on Riemannian manifolds separates cleanly across factors, the canonical geometric median couples them, and its behavior in product spaces has remained largely unexplored. In this paper, we give the first systematic treatment of this problem. After formulating the coupled objective, we establish general existence and uniqueness results: the median is unique on any Hadamard product, and remains locally unique under sharp conditions on curvature and injectivity radius even when one or more factors have positive curvature. We then prove that the estimator enjoys Lipschitz stability to perturbations and the optimal breakdown point, extending classical robustness guarantees to the product-manifold setting. Two practical solvers are proposed, including a Riemannian subgradient method with global sublinear convergence and a product-aware Weiszfeld iteration that achieves local linear convergence when safely away from data singularities. Both algorithms update the factors independently while respecting the latent coupling term, enabling implementation with standard manifold primitives. Simulations on parameter spaces of univariate and multivariate Gaussian distributions endowed with the Bures-Wasserstein geometry show that the median is more resilient to contamination than the Fr'{e}chet mean. The results provide both theoretical foundations and computational tools for robust location inference with heterogeneous manifold-valued data.

Problem

• Study geometric median behavior on product manifolds
• Establish existence and uniqueness of geometric medians
• Develop robust algorithms for manifold-valued data

Innovation

• Formulates coupled objective for geometric medians
• Proposes Riemannian subgradient and Weiszfeld solvers
• Ensures Lipschitz stability and optimal breakdown point