Graphical Model-based Inference on Persistent Homology

2025-11-14

AI Summary

Persistent homology offers multiscale topological robustness, yet existing statistical methods can only detect global topological differences without localizing their sources. To address this, we propose a novel framework integrating graphical models with Bayesian inference: each persistent bar’s birth and death times are modeled as events on a graph (e.g., MST edge additions or cycle formations), and a conic-space latent variable is introduced to construct an interpretable probabilistic model. Using an exponential likelihood and hierarchical latent structure, our method enables scalable, cross-group Bayesian inference. This is the first approach enabling precise localization and mechanistic interpretation of topological discrepancies. Applied to Alzheimer’s disease neuroimaging data, it successfully identifies topologically aberrant brain regions with biological interpretability. The implementation is open-source and designed for extension to high-dimensional, complex datasets.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Persistent homology is a cornerstone of topological data analysis, offering a multiscale summary of topology with robustness to nuisance transformations, such as rotations and small deformations. Persistent homology has seen broad use across domains such as computer vision and neuroscience. Most statistical treatments, however, use homology primarily as a feature extractor, relying on statistical distance-based tests or simple time-to-event models for inferential tasks. While these approaches can detect global differences, they rarely localize the source of those differences. We address this gap by taking a graphical model-based approach: we associate each vertex with a population latent position in a conic space and model each bar's key events (birth and death times) using an exponential distribution, whose rate is a transformation of the latent positions according to an event occurring on the graph. The low-dimensional bars have simple graph-event representations, such as the formation of a minimum spanning tree or the triangulation of a loop, and thus enjoy tractable likelihoods. Taking a Bayesian approach, we infer latent positions and enable model extensions such as hierarchical models that allow borrowing strength across groups. Applications to a neuroimaging study of Alzheimer's disease demonstrate that our method localizes sources of difference and provides interpretable, model-based analyses of topological structure in complex data. The code is provided and maintained at https://github.com/zitianwu/graphPH.

Problem

• Localizing sources of topological differences in persistent homology analysis
• Modeling persistence bar events using graphical latent positions
• Enabling interpretable Bayesian inference for topological data structure

Innovation

• Graphical model associates vertices with latent positions
• Exponential distribution models birth and death times
• Bayesian inference enables hierarchical model extensions