# A Topological Regularizer for Classifiers via Persistent Homology

arXiv: https://arxiv.org/abs/1806.10714 · HTML: https://mlsanity.com/paper/1806.10714/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2018.

## Built on by (most cited first, 30 of 63)

- [Learning with Feature-Dependent Label Noise: A Progressive Approach](https://mlsanity.com/paper/2103.07756/) (2021)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [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)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Cycle Representation Learning for Inductive Relation Prediction](https://mlsanity.com/paper/2110.02510/) (2021)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Neural Approximation of Graph Topological Features](https://mlsanity.com/paper/2201.12032/) (2022)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification](https://mlsanity.com/paper/2306.08805/) (2023)
- [TopoDiffusionNet: A Topology-aware Diffusion Model](https://mlsanity.com/paper/2410.16646/) (2024)
- [Relating Regularization and Generalization through the Intrinsic Dimension of Activations](https://mlsanity.com/paper/2211.13239/) (2022)
- [TopoLiDM: Topology-Aware LiDAR Diffusion Models for Interpretable and Realistic LiDAR Point Cloud Generation](https://mlsanity.com/paper/2507.22454/) (2025)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Self Pre-training with Topology- and Spatiality-aware Masked Autoencoders for 3D Medical Image Segmentation](https://mlsanity.com/paper/2406.10519/) (2024)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [The Flood Complex: Large-Scale Persistent Homology on Millions of Points](https://mlsanity.com/paper/2509.22432/) (2025)
- [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)
- [Topologically penalized regression on manifolds](https://mlsanity.com/paper/2110.13749/) (2021)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [Deep Regression Representation Learning with Topology](https://mlsanity.com/paper/2404.13904/) (2024)
- [Relative Representations: Topological and Geometric Perspectives](https://mlsanity.com/paper/2409.10967/) (2024)
- [Revisiting Point Cloud Completion: Are We Ready For The Real-World?](https://mlsanity.com/paper/2411.17580/) (2024)
- [Efficient Connectivity-Preserving Instance Segmentation with Supervoxel-Based Loss Function](https://mlsanity.com/paper/2501.01022/) (2025)

## Abstract (arXiv)

Regularization plays a crucial role in supervised learning. Most existing methods enforce a global regularization in a structure agnostic manner. In this paper, we initiate a new direction and propose to enforce the structural simplicity of the classification boundary by regularizing over its topological complexity. In particular, our measurement of topological complexity incorporates the importance of topological features (e.g., connected components, handles, and so on) in a meaningful manner, and provides a direct control over spurious topological structures. We incorporate the new measurement as a topological penalty in training classifiers. We also pro- pose an efficient algorithm to compute the gradient of such penalty. Our method pro- vides a novel way to topologically simplify the global structure of the model, without having to sacrifice too much of the flexibility of the model. We demonstrate the effectiveness of our new topological regularizer on a range of synthetic and real-world datasets.