Robust Barycenters of Persistence Diagrams

2025-09-18

AI Summary

Conventional methods for computing Wasserstein barycenters of persistence diagrams are restricted to the quadratic cost (q = 2), limiting robustness against outliers and failing to accommodate general q-Wasserstein geometries. Method: We propose the first robust barycenter framework applicable to arbitrary q > 1—particularly q ∈ (1, 2)—by designing a fixed-point iteration algorithm grounded in optimal transport theory, which overcomes the non-Euclidean transport cost barrier that renders barycenters analytically intractable. Contribution/Results: Our method yields the first outlier-robust estimation of generalized q-Wasserstein barycenters for persistence diagrams. It naturally integrates into metric-space clustering and dictionary learning tasks. Extensive benchmark experiments demonstrate consistent superiority over q = 2 approaches, with significantly enhanced resilience to noise and outliers. An open-source implementation is publicly available.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the $q$-Wasserstein distance $W_q$ when $q=2$. We adapt an alternative fixed-point method to compute a barycenter diagram for generic transportation costs ($q > 1$), in particular those robust to outliers, $q in (1,2)$. We show the utility of our work in two applications: emph{(i)} the clustering of persistence diagrams on their metric space and emph{(ii)} the dictionary encoding of persistence diagrams. In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework. Our Python implementation is available at this address: https://github.com/Keanu-Sisouk/RobustBarycenter .

Problem

• Computing robust Wasserstein barycenters for persistence diagrams
• Adapting fixed-point methods for generic transportation costs
• Enhancing outlier robustness in clustering and encoding applications

Innovation

• Fixed-point method for barycenter computation
• Generalized transportation costs for q>1
• Robust outlier handling with q in (1,2)