Advancing Topological Data Analysis

Bridging between theory and applications

Topological data analysis (TDA) brings mathematical techniques from algebraic topology to the applied domains, with applications in biochemistry, cosmology, materials science, neuroscience, and more. This project, titled “Efficient Computation of Generalized Persistence Diagrams,” aims to improve upon TDA methods by correlating multiple measurements to enable more powerful analyses and a broader array of applications.

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The Problem

TDA emphasizes methods that are stable (making them resilient to noise in the data), and computationally efficient (making them practical across a range of applications). However, one significant limitation that practitioners encounter is that one of the main methods in TDA, persistent homology, is limited to studying 1-parameter data (measurements of a single quantity). This represents a missed opportunity, because in practice, multiple measurements are available, and the correlation between them reveals important features of the problem.

Recently, researchers introduced generalized persistent homology, which interprets persistence as a Möbius inversion of a certain function derived from the changes in topology of the data as one or multiple parameters are varied. This construction generalizes all the properties of 1-parameter persistence needed in applications, including stability and the particular structure of the diagrams used in machine learning and statistical pipelines. However, additional work is needed to move this approach from theory to practice.

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How We’re Addressing It

In this project, we are developing a software implementation for computing generalized persistence diagrams to bridge between theory and applications. In our preliminary work, we introduced an efficient output-sensitive algorithm for computing generalized persistence diagrams of 2-parameter filtrations. In addition to extending this work through software for practitioners, we are training graduate students in advanced TDA techniques.

Project Team

Outcomes

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