On the Limits of Topological Data Analysis for Statistical Inference

TL;DR

This paper explores the potential and limitations of using topological data analysis for statistical inference, identifying conditions for validity and providing examples of models that are invariant under topological summaries.

Contribution

It establishes necessary and sufficient conditions for valid statistical inference using topological summaries and demonstrates model invariance properties.

Findings

Identifies conditions for valid inference with topological summaries

Provides models invariant under topological summaries

Highlights limitations of topological data analysis in inference

Abstract

Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data arises from complex sources. In this paper, we examine the use of topological summary statistics through the lens of statistical inference. We investigate necessary and sufficient conditions under which \textit{valid statistical inference} is possible using {topological summary statistics}. Additionally, we provide examples of models that demonstrate invariance with respect to topological summaries.