The Density of Cross-Persistence Diagrams and Its Applications

TL;DR

This paper systematically studies the density of cross-persistence diagrams in Topological Data Analysis, establishing theoretical foundations, developing a machine learning framework for density prediction, and demonstrating improved data analysis capabilities including noise utility.

Contribution

It introduces the first theoretical and machine learning framework for analyzing the density of cross-persistence diagrams, enhancing the understanding and application of TDA in data analysis.

Findings

Density of cross-persistence diagrams exists and can be theoretically characterized.

Machine learning models can predict cross-persistence density from point cloud data.

Adding noise can improve the ability to distinguish different point clouds.

Abstract

Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing individual manifolds, persistence diagrams do not account for interactions between pairs of them. Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds. In this work, we present the first systematic study of the density of cross-persistence diagrams. We prove its existence, establish theoretical foundations for its statistical use, and design the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices. Our statistical approach enables the distinction of point clouds sampled from different manifolds by leveraging the linear characteristics of cross-persistence diagrams. Interestingly, we find that introducing noise can enhance our ability to distinguish point clouds, uncovering its novel utility in TDA applications. We demonstrate the effectiveness of our methods through experiments on diverse datasets, where our approach consistently outperforms existing techniques in density prediction and achieves superior results in point cloud distinction tasks. Our findings contribute to a broader understanding of cross-persistence diagrams and open new avenues for their application in data analysis, including potential insights into time-series domain tasks and the geometry of AI-generated texts. Our code is publicly available at https://github.com/Verdangeta/TDA_experiments