# A Topology Layer for Machine Learning

arXiv: https://arxiv.org/abs/1905.12200 · HTML: https://mlsanity.com/paper/1905.12200/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2019.

## Built on by (most cited first, 30 of 80)

- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (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)
- [Artificial Text Detection via Examining the Topology of Attention Maps](https://mlsanity.com/paper/2109.04825/) (2021)
- [A persistent homology-based topological loss for CNN-based multi-class segmentation of CMR](https://mlsanity.com/paper/2107.12689/) (2021)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Model](https://mlsanity.com/paper/2412.06011/) (2024)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [Persistent Homology with Improved Locality Information for more Effective Delineation](https://mlsanity.com/paper/2110.06295/) (2021)
- [TopoDiffusionNet: A Topology-aware Diffusion Model](https://mlsanity.com/paper/2410.16646/) (2024)
- [Cycle Registration in Persistent Homology with Applications in Topological Bootstrap](https://mlsanity.com/paper/2101.00698/) (2021)
- [TopoLiDM: Topology-Aware LiDAR Diffusion Models for Interpretable and Realistic LiDAR Point Cloud Generation](https://mlsanity.com/paper/2507.22454/) (2025)
- [Augmenting Molecular Deep Generative Models with Topological Data Analysis Representations](https://mlsanity.com/paper/2106.04464/) (2021)
- [Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integrity](https://mlsanity.com/paper/2412.16619/) (2024)
- [Topological Deep Learning: Classification Neural Networks](https://mlsanity.com/paper/2102.08354/) (2021)
- [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)
- [BERTops: Studying BERT Representations under a Topological Lens](https://mlsanity.com/paper/2205.00953/) (2022)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology](https://mlsanity.com/paper/2505.20435/) (2025)
- [Rethinking Persistent Homology for Visual Recognition](https://mlsanity.com/paper/2207.04220/) (2022)
- [Learning Persistent Community Structures in Dynamic Networks via Topological Data Analysis](https://mlsanity.com/paper/2401.03194/) (2024)
- [Revisiting Point Cloud Completion: Are We Ready For The Real-World?](https://mlsanity.com/paper/2411.17580/) (2024)
- [STITCH: Surface reconstrucTion using Implicit neural representations with Topology Constraints and persistent Homology](https://mlsanity.com/paper/2412.18696/) (2024)
- [A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.02836/) (2026)

## Abstract (arXiv)

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: the topological layer can (i) regularize data reconstruction or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained with persistence features. The code (www.github.com/bruel-gabrielsson/TopologyLayer) is publicly available and we hope its availability will facilitate the use of persistent homology in deep learning and other gradient based applications.