Persistence-Augmented Neural Networks

Elena Xinyi Wang; Arnur Nigmetov; Dmitriy Morozov

cs.LGApr 9, 2026

Persistence-Augmented Neural Networks

Authors: Elena Xinyi Wang, Arnur Nigmetov, Dmitriy Morozov

Organizations: Department of Informatics, University of Fribourg · Lawrence Berkeley National Laboratory

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.

Figures & tables

Figure 1: Dual graph and simplification

Figure 2: Construction of the hierarchical graph input for GNNs. Each layer represents a simplification level

\mathcal{V}_{\ell}

, where nodes correspond to Morse–Smale regions defined by

(m,M)

pairs. Intra-level edges (dashed lines) connect adjacent regions, while inter-level edges (solid lines) indicate merge relationships as simplification progresses.

Figure 3: Visualization of different levels of simplification

Model	Accuracy (%)	Cohen’s Kappa (%)
CNN (Original)	84.0 $\pm$ 0.7	71.0 $\pm$ 0.6
CNN + Persistence Image	88.5 $\pm$ 0.5	80.2 $\pm$ 0.8
CNN + Persistence Landscape	89.1 $\pm$ 0.6	82.3 $\pm$ 0.9
CNN + Persistence Augmentation	95.0 $\pm$ 0.5	91.0 $\pm$ 0.6
GNN (Full Hierarchy)	92.0 $\pm$ 1.1	89.0 $\pm$ 1.0
GNN (w/o Base Layer)	90.5 $\pm$ 1.0	86.5 $\pm$ 1.3

Table 1: Classification performance on the histopathology dataset. Results are reported as mean accuracy / Cohen’s kappa with standard deviation over 5-fold cross-validation.

Model	Porosity 0.6	Porosity 0.7	Porosity 0.75
CNN (Original)	0.74 $\pm$ 0.04	0.28 $\pm$ 0.06	0.88 $\pm$ 0.02
CNN + Persistence Image	0.80 $\pm$ 0.03	0.53 $\pm$ 0.05	0.90 $\pm$ 0.01
CNN + Persistence Landscape	0.81 $\pm$ 0.03	0.56 $\pm$ 0.04	0.91 $\pm$ 0.01
CNN + Persistence Augmentation	0.88 $\pm$ 0.02	0.72 $\pm$ 0.03	0.95 $\pm$ 0.01
GNN (Full Hierarchy)	0.86 $\pm$ 0.03	0.76 $\pm$ 0.04	0.92 $\pm$ 0.01
GNN (w/o Base Layer)	0.84 $\pm$ 0.03	0.70 $\pm$ 0.04	0.91 $\pm$ 0.01

Table 2: Regression performance on the porous material dataset. Results are reported as mean test

R^{2}

scores

\pm

standard deviation over 5-fold cross-validation.

Data Augmentation

Data augmentation expands or transforms training examples to improve learning, data efficiency, and robustness to distribution shifts.

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How Data Augmentation Shapes Neural Representations

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Neural Representation Geometry Data Augmentation

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Data augmentation is widely recognized for improving generalization in deep networks, yet its impact on the geometry of learned representations remains poorly understood. In this work, we characterize how different data augmentation strategies reshape internal representations in neural networks. Using tools from shape analysis, we embed network hidden representations into a metric space where distance is invariant to scaling, translation, rotation and reflection. We show that increasing augmentation strength leads to well-behaved trajectories in this space, and that different augmentation types steer representations in distinct directions. Moreover, we investigate how neural representation shapes are distorted along data augmentation trajectories, and show that insights from neural geometry can predict which representations provide the most improvement when ensembling models. Our results reveal shared geometric patterns across architectures and seeds, and suggest that analyzing shape-space trajectories offers a principled tool for understanding and comparing data augmentation methods.

Abstract

Data Augmentation

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