用代数拓扑与图像处理生成高维多样拓扑数据，助力神经网络学习复杂结构
Generating Topologically and Geometrically Diverse Manifold Data in Dimensions Four and Below
- 结合代数拓扑与形态学方法生成2-4维拓扑数据
- 可为4D卷积网络提供带拓扑标签的合成数据
- 适合研究拓扑感知深度学习的学者参考
理解数据的拓扑特性对多个研究领域至关重要。近期研究表明，合成4维图像型数据有助于训练4维卷积神经网络以识别其中的拓扑特征，且这类模型对图像预处理技术具有更强鲁棒性，而传统拓扑数据分析方法如持久同调（persistent homology）则难以应对。本文探讨如何利用代数拓扑方法，并结合形态学等图像处理技术，在二维和三维中生成拓扑复杂且多样的图像型数据，同时附带拓扑标签。这些方法为在四维中实现类似目标提供了路线图。
原文摘要 · Abstract (English)
Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological features in these data. These models also appear to tolerate the use of image preprocessing techniques where existing topological data analysis techniques such as persistent homology do not. This paper investigates how methods from algebraic topology, combined with image processing techniques such as morphology, can be used to generate topologically sophisticated and diverse-looking 2-, 3-, and 4D image-type data with topological labels in simulation. These approaches are illustrated in 2D and 3D with the aim of providing a roadmap towards achieving this in 4D.
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