# On the Convergence of Optimizing Persistent-Homology-Based Losses

arXiv: https://arxiv.org/abs/2206.02946 · HTML: https://mlsanity.com/paper/2206.02946/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 37 references that are in the map.
- First posted on arXiv 6 Jun 2022.

## Builds on (most cited first, 18 of 18)

- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [A Topological Filter for Learning with Label Noise](https://mlsanity.com/paper/2012.04835/) (2020)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Learning metrics for persistence-based summaries and applications for graph classification](https://mlsanity.com/paper/1904.12189/) (2019)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Trigger Hunting with a Topological Prior for Trojan Detection](https://mlsanity.com/paper/2110.08335/) (2021)
- [Link Prediction with Persistent Homology: An Interactive View](https://mlsanity.com/paper/2102.10255/) (2021)
- [Topological Detection of Trojaned Neural Networks](https://mlsanity.com/paper/2106.06469/) (2021)
- [A Fast and Robust Method for Global Topological Functional Optimization](https://mlsanity.com/paper/2009.08496/) (2020)
- [TopoTxR: A Topological Biomarker for Predicting Treatment Response in Breast Cancer](https://mlsanity.com/paper/2105.06049/) (2021)
- [Topologically Densified Distributions](https://mlsanity.com/paper/2002.04805/) (2020)

## Built on by (most cited first, 2 of 2)

- [Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integrity](https://mlsanity.com/paper/2412.16619/) (2024)
- [Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximation](https://mlsanity.com/paper/2509.06694/) (2025)

## Abstract (arXiv)

Topological loss based on persistent homology has shown promise in various applications. A topological loss enforces the model to achieve certain desired topological property. Despite its empirical success, less is known about the optimization behavior of the loss. In fact, the topological loss involves combinatorial configurations that may oscillate during optimization. In this paper, we introduce a general purpose regularized topology-aware loss. We propose a novel regularization term and also modify existing topological loss. These contributions lead to a new loss function that not only enforces the model to have desired topological behavior, but also achieves satisfying convergence behavior. Our main theoretical result guarantees that the loss can be optimized efficiently, under mild assumptions.