Persistence-Augmented Neural Networks

TL;DR

This paper introduces a persistence-based data augmentation method that encodes local topological features to improve deep learning models on complex data, demonstrating efficiency and superior performance.

Contribution

It presents a novel, efficient topological augmentation framework compatible with neural networks, capturing local geometric structures across multiple scales.

Findings

Outperforms global TDA descriptors like persistence images and landscapes.

Maintains performance while reducing memory by pruning the hierarchy.

Efficient with $O(n \,\log\, n)$ complexity, suitable for large datasets.

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.