Persistence-based topological optimization: a survey

2026-03-24

AI Summary

This work addresses the ongoing challenge of effectively incorporating topological priors into optimization problems. It proposes a systematic framework for topological optimization grounded in persistent homology, which unifies gradient-based optimization with a differentiable topological regularization loss to enable end-to-end learning of topological features. Providing a comprehensive survey of theoretical and algorithmic advances over the past decade, the paper offers—for the first time—an accessible, unified introduction tailored to mathematicians and data scientists new to the field. Accompanied by an open-source library, this contribution aims to lower entry barriers and foster broader adoption of topological data analysis in machine learning and data science communities.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Machine learning and data science for the Web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• persistent homology
• topological optimization
• gradient-based optimization
• topological data analysis
• persistence-based loss

Innovation

• persistent homology
• topological optimization
• gradient-based optimization
• topological data analysis
• open-source library