Statistical Analysis for Topological Data Analysis and Application to Machine Learning

Abstract

This presentation provides a comprehensive overview of Topological Data Analysis (TDA), covering its fundamental concepts, methods for statistical analysis, and diverse applications within the field of machine learning. TDA is an emerging field that characterizes the underlying geometric structure and "shape" of high-dimensional data, by extracting inherent topological features. The core technique, Persistent Homology, identifies robust topological patterns-—such as clusters and holes of various dimensions—-that persist across multiple scales of observation. By capturing these multiscale patterns, TDA provides unique structural insights that complement traditional statistical descriptors and enhance the performance of machine learning models. The first part of this talk addresses the statistical challenges inherent in TDA. Since topological features are extracted from finite data samples, they are naturally subject to sampling variability. After briefly introducing basic concepts of persistent homology, I will discuss how to quantify the uncertainty of these extracted features and introduce the use of confidence sets to distinguish between "true" topological signals and "noise" arising from randomness in the data. This framework allows for rigorous statistical inference within the topological domain. The second part explores two primary paradigms for integrating TDA into machine learning workflows. The first is featurization, where complex topological summaries are transformed into vector-valued or functional representations suitable for standard learning algorithms. The second is evaluation, which uses topological metrics to assess the structural integrity of generated data or the performance of predictive models. Through various examples, I will demonstrate how these approaches reveal the transformative potential of TDA in machine learning.

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