用拓扑方法增强神经网络，让模型更懂数据的局部形状结构。
Persistence-Augmented Neural Networks
- 基于莫尔斯-斯梅尔复形，捕捉梯度流区域的层次演化。
- 在病理图像和多孔材料预测中，性能优于传统拓扑描述符。
- 计算高效（O(n log n)），适合大规模数据，可剪枝降内存。
拓扑数据分析（TDA）能描述数据的形状特征，但将其融入深度学习仍具挑战，尤其在保留局部几何结构而非全局摘要时。本文提出一种基于持续性（persistence）的数据增强框架，利用莫尔斯-斯梅尔复形编码局部梯度流区域及其层次演化。该表示兼容卷积与图神经网络，可在多尺度下保持空间局部拓扑信息。重要的是，增强过程计算复杂度仅为O(n log n)，适用于大规模数据集。我们在组织病理学图像分类与3D多孔材料回归任务上评估，结果表明该方法持续优于基线及全局TDA描述符（如持久性图像与景观）。此外，剪枝层级基础可降低内存占用，同时保持竞争力。这些结果凸显了局部、结构化拓扑增强在跨模态可扩展且可解释学习中的潜力。
原文摘要 · Abstract (English)
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.
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