Persistence-Augmented Neural Networks

2026-04-09

AI Summary

This work addresses the limitation of existing deep learning approaches that predominantly rely on global topological summaries and thus struggle to effectively exploit local structural information. The authors propose a novel persistent homology–based data augmentation framework that, for the first time, integrates Morse–Smale complexes into neural networks to efficiently and scalably encode local gradient flow regions along with their multiscale hierarchical evolution. The method is compatible with both convolutional and graph neural architectures and enables a memory-efficient hierarchical pruning strategy. Evaluated on histopathology image classification and 3D porous material regression tasks, the approach significantly outperforms baseline models and conventional global topological data analysis descriptors, simultaneously enhancing model interpretability and computational efficiency.

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.

Problem

• Topological Data Analysis
• local geometric structure
• persistence
• Morse-Smale complex
• deep learning

Innovation

• Persistence-Augmented Neural Networks
• Morse-Smale Complex
• Topological Data Analysis
• Local Topological Features
• Scalable Data Augmentation