Graphical Model-based Inference on Persistent Homology

TL;DR

This paper introduces a graphical model-based Bayesian method for analyzing persistent homology, enabling localization of differences and interpretability in complex data like neuroimaging, surpassing traditional global difference detection methods.

Contribution

It proposes a novel Bayesian graphical model approach for persistent homology that localizes differences and allows hierarchical extensions, improving interpretability over existing methods.

Findings

Successfully localized sources of differences in neuroimaging data.

Provided interpretable, model-based analysis of topological structures.

Demonstrated effectiveness in Alzheimer's disease study.

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/zitian-wu/graphPH.