MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology

2025-10-16

AI Summary

This work addresses the challenge of analyzing non-hierarchical multiscale clustering structures. We propose Multiscale Clustering Bifiltration (MCbiF), the first framework to integrate two-parameter persistent homology into multiscale clustering modeling. MCbiF constructs an abstract simplicial bifiltration that simultaneously captures, across resolution scales, the refinement relations among cluster partitions (via zero-dimensional homology) and higher-order structural inconsistencies (via one-dimensional homology). Topological features are stably and decomposably represented using the Hilbert function. Empirically, MCbiF significantly outperforms information-theoretic baselines on classification and regression tasks. Moreover, it enables the first non-hierarchical, quantitative analysis of dynamic social group structures in wild mice. By unifying multiscale clustering with multiparameter topological data analysis, MCbiF establishes the first general-purpose, topology-driven modeling paradigm for multiscale clustering.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF feature maps outperform information-based baseline features on both regression and classification tasks on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time.

Problem

• Measuring topological autocorrelation in multiscale clusterings
• Detecting violations of refinement order across partition scales
• Capturing higher-order cluster inconsistencies via persistent homology

Innovation

• Uses 2-parameter persistent homology for multiscale clustering analysis
• Encodes cluster intersection patterns via abstract simplicial complexes
• Derives topological feature maps from stable Hilbert functions