# Persistence-based topological optimization: a survey

arXiv: https://arxiv.org/abs/2603.24613 · HTML: https://mlsanity.com/paper/2603.24613/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 138 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Builds on (most cited first, 5 of 5)

- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Statistical Analysis of Persistence Intensity Functions](https://mlsanity.com/paper/1510.02502/) (2015)
- [Applying Topological Persistence in Convolutional Neural Network for Music Audio Signals](https://mlsanity.com/paper/1608.07373/) (2016)
- [Medians of populations of persistence diagrams](https://mlsanity.com/paper/1307.8300/) (2013)
- [Geodesics in persistence diagram space](https://mlsanity.com/paper/1905.10820/) (2019)

## Built on by (most cited first, 1 of 1)

- [Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learning](https://mlsanity.com/paper/2604.26370/) (2026)

## Abstract (arXiv)

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.