# Persistence-Augmented Neural Networks

arXiv: https://arxiv.org/abs/2604.08469 · HTML: https://mlsanity.com/paper/2604.08469/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Clique topology reveals intrinsic geometric structure in neural correlations](https://mlsanity.com/paper/1502.06172/) (2015)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [Topological analysis of the connectome of digital reconstructions of neural microcircuits](https://mlsanity.com/paper/1601.01580/) (2016)
- [Equivariant geometric learning for digital rock physics: estimating formation factor and effective permeability tensors from Morse graph](https://mlsanity.com/paper/2104.05608/) (2021)
- [Approximating Continuous Functions on Persistence Diagrams Using Template Functions](https://mlsanity.com/paper/1902.07190/) (2019)
- [Topological Descriptors Help Predict Guest Adsorption in Nanoporous Materials](https://mlsanity.com/paper/2001.05972/) (2020)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)

## Abstract (arXiv)

Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than summarizing it globally. We propose a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse-Smale complex. This representation, compatible with both convolutional and graph neural networks, retains spatially localized topological information across multiple scales. Importantly, the augmentation procedure itself is efficient, with computational complexity O(n log n), making it practical for large datasets. We evaluate our method on histopathology image classification and 3D porous material regression, where it consistently outperforms baselines and global TDA descriptors such as persistence images and landscapes. We also show that pruning the base level of the hierarchy reduces memory usage while maintaining competitive performance. These results highlight the potential of local, structured topological augmentation for scalable and interpretable learning across data modalities.