Persistence-Augmented Neural Networks
2026-04-09 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce a new method that uses topological features—ways to capture the shapes and structures in data—locally instead of just summarizing them globally. Their technique uses a tool called the Morse-Smale complex to encode detailed, multi-scale information about data regions, and it works well with common deep learning models like convolutional and graph neural networks. They show their approach is efficient for large datasets and improves performance on tasks involving medical images and 3D materials compared to existing global methods. Additionally, they demonstrate that simplifying parts of their representation can save memory without greatly hurting results.
Topological Data AnalysisMorse-Smale complexPersistence-based augmentationLocal gradient flowConvolutional Neural NetworksGraph Neural NetworksPersistence imagesPersistence landscapesHistopathology3D porous materials
Authors
Elena Xinyi Wang, Arnur Nigmetov, Dmitriy Morozov
Abstract
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.