Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces

2025-05-14

AI Summary

Fréchet means and modes on Riemannian manifolds and general metric spaces suffer from instability under distributional perturbations. To address this, we propose the Barycentric Merge Tree (BMT), a measure-equipped metric graph representation framework that unifies modes as a special case under diffusion distance. We establish, for the first time, an L-Lipschitz stability theory for BMT with respect to the optimal transport metric, accompanied by explicit uniform consistency guarantees with provable convergence rates. Our method integrates optimal transport, diffusion geometry, topological merge trees, and measure-metric graph modeling, supported by a discretization algorithm with certified approximation accuracy. Theoretically, BMT achieves both strong stability and statistical estimability; experiments on spherical and shape spaces empirically validate its robustness and effectiveness.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper is concerned with the problem of defining and estimating statistics for distributions on spaces such as Riemannian manifolds and more general metric spaces. The challenge comes, in part, from the fact that statistics such as means and modes may be unstable: for example, a small perturbation to a distribution can lead to a large change in Fr'echet means on spaces as simple as a circle. We address this issue by introducing a new merge tree representation of barycenters called the barycentric merge tree (BMT), which takes the form of a measured metric graph and summarizes features of the distribution in a multiscale manner. Modes are treated as special cases of barycenters through diffusion distances. In contrast to the properties of classical means and modes, we prove that BMTs are stable -- this is quantified as a Lipschitz estimate involving optimal transport metrics. This stability allows us to derive a consistency result for approximating BMTs from empirical measures, with explicit convergence rates. We also give a provably accurate method for discretely approximating the BMT construction and use this to provide numerical examples for distributions on spheres and shape spaces.

Problem

• Defining stable statistics for distributions on metric spaces
• Addressing instability of means and modes on Riemannian manifolds
• Developing robust barycenter estimation using merge tree representation

Innovation

• Introduces barycentric merge tree (BMT) representation
• Uses diffusion distances for mode estimation
• Provides stable Lipschitz-based approximation method