# A Fast and Robust Method for Global Topological Functional Optimization

arXiv: https://arxiv.org/abs/2009.08496 · HTML: https://mlsanity.com/paper/2009.08496/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2020.

## Built on by (most cited first, 15 of 15)

- [Artificial Text Detection via Examining the Topology of Attention Maps](https://mlsanity.com/paper/2109.04825/) (2021)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Robust Lipschitz Bandits to Adversarial Corruptions](https://mlsanity.com/paper/2305.18543/) (2023)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Topologically Regularized Data Embeddings](https://mlsanity.com/paper/2110.09193/) (2021)
- [Optimisation of Spectral Wavelets for Persistence-based Graph Classification](https://mlsanity.com/paper/2101.05201/) (2021)
- [On the Convergence of Optimizing Persistent-Homology-Based Losses](https://mlsanity.com/paper/2206.02946/) (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)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [Topologically Regularized Data Embeddings](https://mlsanity.com/paper/2301.03338/) (2023)
- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)

## Abstract (arXiv)

Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.