Persistence-Augmented Neural Networks
This work addresses the problem of scalable and interpretable learning across data modalities for researchers and practitioners in fields like medical imaging and materials science, though it is incremental as it builds on existing TDA methods.
The paper tackles the challenge of integrating local topological features into deep learning by proposing a persistence-based data augmentation framework using the Morse-Smale complex, which outperforms baselines and global TDA descriptors in tasks like histopathology image classification and 3D porous material regression with efficient O(n log n) complexity.
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.