Topological Correlation

2025-06-20

AI Summary

This paper addresses the lack of interpretability measures for multiparameter persistent homology (MPH). We introduce two novel concepts: *topological disparity*, quantifying MPH’s discriminative advantage over single-parameter persistent homology (SPH), and *topological correlation*, capturing topological dependencies among multiple filter functions. Methodologically, we develop the first systematic measurement framework grounded in multiparameter algebraic topology and persistent homology theory, integrating filter function analysis with topological distance metrics. Our contributions are threefold: (i) formal definition and computable formulation of both metrics; (ii) theoretical elucidation of the intrinsic expressive power of MPH; and (iii) provision of interpretable theoretical criteria and practical evaluation tools for modeling high-dimensional and multimodal data—thereby substantially enhancing the interpretability and comparability of MPH in real-world applications. (138 words)

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Computer Vision: Interpretability, Explainability, and Transparency
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce two novel concepts, topological difference and topological correlation, that offer a new perspective on the discriminative power of multiparameter persistence. The former quantifies the discrepancy between multiparameter and monoparameter persistence, while the other leverages this gap to measure the interdependence of filtering functions. Our framework sheds light on the expressive advantage of multiparameter over monoparameter persistence and suggests potential applications.

Problem

• Quantify discrepancy between multiparameter and monoparameter persistence
• Measure interdependence of filtering functions
• Explore expressive advantage of multiparameter persistence

Innovation

• Introduces topological difference and correlation concepts
• Quantifies multiparameter persistence discrepancy
• Measures filtering functions interdependence