将微分拓扑融入深度学习，提升医学图像识别效果
Manifold Topological Deep Learning for Biomedical Data
- 用霍奇理论分解图像向量场为三类正交分量
- 在MedMNIST v2上准确率显著超越现有方法
- 适合处理图像等流形数据的深度学习研究者
近期，拓扑深度学习（TDL）将代数拓扑与深度神经网络结合，在点云数据处理中取得显著进展，成为数据科学的新范式。然而，由于微分拓扑的挑战，TDL尚未应用于可微流形数据（如图像）。本文首次提出流形拓扑深度学习（MTDL），利用霍奇理论揭示微分拓扑的潜力，将原始图像表示为带向量场的光滑流形，并基于霍奇分解将其分解为三个正交分量，再拼接为CNN输入。在包含717,287张生物医学图像的MedMNIST v2基准数据库（涵盖11个2D和6个3D数据集）上，MTDL显著优于其他对比方法，将TDL拓展至光滑流形数据的广泛场景。
原文摘要 · Abstract (English)
Recently, topological deep learning (TDL), which integrates algebraic topology with deep neural networks, has achieved tremendous success in processing point-cloud data, emerging as a promising paradigm in data science. However, TDL has not been developed for data on differentiable manifolds, including images, due to the challenges posed by differential topology. We address this challenge by introducing manifold topological deep learning (MTDL) for the first time. To highlight the power of Hodge theory rooted in differential topology, we consider a simple convolutional neural network (CNN) in MTDL. In this novel framework, original images are represented as smooth manifolds with vector fields that are decomposed into three orthogonal components based on Hodge theory. These components are then concatenated to form an input image for the CNN architecture. The performance of MTDL is evaluated using the MedMNIST v2 benchmark database, which comprises 717,287 biomedical images from eleven 2D and six 3D datasets. MTDL significantly outperforms other competing methods, extending TDL to a wide range of data on smooth manifolds.
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