Stability for Inference with Persistent Homology Rank Functions

TL;DR

This paper explores the use of persistent homology rank functions in topological data analysis, establishing their stability and demonstrating their effectiveness in statistical and machine learning applications with complex data.

Contribution

It introduces stability results for persistent homology rank functions and evaluates their performance in statistical inference and machine learning tasks.

Findings

Rank functions are stable under suitable metrics.

Rank functions improve inference and learning over non-persistence methods.

Application to real data shows enhanced performance.

Abstract

Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical settings is challenging due to their complex geometric structure. In this paper, we revisit the persistent homology rank function, which is mathematically equivalent to a barcode and persistence diagram, as a tool for statistics and machine learning. Rank functions, being functions, enable the direct application of the statistical theory of functional data analysis (FDA)-a domain of statistics adapted for data in the form of functions. A key challenge they present over barcodes in practice, however, is their lack of stability-a property that is crucial to validate their use as a faithful representation of the data and therefore a viable summary statistic. In this paper, we fill this gap by deriving two stability results for persistent homology rank functions under a suitable metric for FDA integration. We then study the performance of rank functions in functional inferential statistics and machine learning on real data applications, in both single and multiparameter persistent homology. We find that the use of persistent homology captured by rank functions offers a clear improvement over existing non-persistence-based approaches.