Fast Topological Signal Identification and Persistent Cohomological Cycle Matching

TL;DR

This paper introduces a fast, cohomology-based method for identifying significant topological features and matching cycles across datasets in topological data analysis, enabling scalable analysis of large, complex data.

Contribution

It develops a general, efficient approach for topological signal detection and cycle matching using cohomology, extending to non-Morse filtrations and large datasets.

Findings

Achieves cycle matching on thousands of points in minutes

Extends methods to non-Morse filtrations

Demonstrates scalability on real-world datasets

Abstract

Within the context of topological data analysis, the problems of identifying topological significance and matching signals across datasets are important and useful inferential tasks in many applications. The limitation of existing solutions to these problems, however, is computational speed. In this paper, we harness the state-of-the-art for persistent homology computation by studying the problem of determining topological prevalence and cycle matching using a cohomological approach, which increases their feasibility and applicability to a wider variety of applications and contexts. We demonstrate this on a wide range of real-life, large-scale, and complex datasets. We extend existing notions of topological prevalence and cycle matching to include general non-Morse filtrations. This provides the most general and flexible state-of-the-art adaptation of topological signal identification and persistent cycle matching, which performs comparisons of orders of ten for thousands of sampled points in a matter of minutes on standard institutional HPC CPU facilities.