Persistence-based topological optimization: a survey

TL;DR

This survey reviews the development of persistence-based topological optimization, covering theoretical foundations, algorithms, and applications, and provides an open-source library for practical experimentation.

Contribution

It offers a comprehensive overview of optimization techniques for persistence-based topological descriptors, including theoretical insights, algorithmic methods, and practical tools.

Findings

Various gradient-based optimization techniques have been developed for persistence-based losses.

The survey introduces an accessible framework for newcomers to understand topological optimization.

An open-source library is provided to facilitate research and application in the field.

Abstract

Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a way to incorporate topological priors or to regularize machine learning models. This is usually achieved by minimizing adequate, topologically-informed losses based on these descriptors, which, in turn, naturally raises theoretical and practical questions about the possibility of optimizing such loss functions using gradient-based algorithms. This has been an active research field in the topological data analysis community over the last decade, and various techniques have been developed to enable optimization of persistence-based loss functions with gradient descent schemes. This survey presents the current state of this field, covering its theoretical foundations, the algorithmic aspects, and showcasing practical uses in several applications. It includes a detailed introduction to persistence theory and, as such, aims at being accessible to mathematicians and data scientists newcomers to the field. It is accompanied by an open-source library which implements the different approaches covered in this survey, providing a convenient playground for researchers to get familiar with the field.