# Topological Optimization with Big Steps

arXiv: https://arxiv.org/abs/2203.16748 · HTML: https://mlsanity.com/paper/2203.16748/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Mar 2022.

## Built on by (most cited first, 16 of 16)

- [Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation](https://mlsanity.com/paper/2411.03228/) (2024)
- [Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology](https://mlsanity.com/paper/2407.04683/) (2024)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Spherical Rotation Dimension Reduction with Geometric Loss Functions](https://mlsanity.com/paper/2204.10975/) (2022)
- [Nonparametric Multi-shape Modeling with Uncertainty Quantification](https://mlsanity.com/paper/2206.09127/) (2022)
- [A Practical Solver for Scalar Data Topological Simplification](https://mlsanity.com/paper/2407.12399/) (2024)
- [Topological Parallax: A Geometric Specification for Deep Perception Models](https://mlsanity.com/paper/2306.11835/) (2023)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data](https://mlsanity.com/paper/2502.16049/) (2025)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)
- [Persistence-Augmented Neural Networks](https://mlsanity.com/paper/2604.08469/) (2026)
- [Sparse cubical complexes for efficient topology-preservation in image data](https://mlsanity.com/paper/2609.37177/) (2026)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)
- [Operational Abstractions of Neural Network Concepts via Topological Representations](https://mlsanity.com/paper/2610.05017/) (2026)

## Abstract (arXiv)

Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. This relies on another contribution of this paper, which eliminates the need to examine a factorial number of permutations of simplices with the same value. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude.