Hypothesis Testing for Shapes using Vectorized Persistence Diagrams

TL;DR

This paper introduces a two-stage hypothesis testing method for vectorized persistence diagrams in topological data analysis, improving inference accuracy on data shape across diverse datasets.

Contribution

It proposes a novel two-stage hypothesis test that enhances power and controls false discovery rates for analyzing shapes with persistent homology.

Findings

Improved accuracy over existing methods

Effective on simulated and real-world data

Controlled false discovery rates

Abstract

Topological data analysis involves the statistical characterization of the shape of data. Persistent homology is a primary tool of topological data analysis, which can be used to analyze topological features and perform statistical inference. In this paper, we present a two-stage hypothesis test for vectorized persistence diagrams. The first stage filters vector elements in the vectorized persistence diagrams to enhance the power of the test. The second stage consists of multiple hypothesis tests, with false positives controlled by false discovery rates. We demonstrate the flexibility of our method by applying it to a variety of simulated and real-world data types. Our results show that the proposed hypothesis test enables accurate and informative inferences on the shape of data compared to the existing hypothesis testing methods for persistent homology.