用康托集构造神经网络，测试几何与拓扑复杂度度量
CantorNet: A Sandbox for Testing Geometrical and Topological Complexity Measures
- 基于康托集构造分形神经网络，可控制复杂度变化
- 决策边界任意锯齿状但解析已知，支持精确分析
- 适合研究复杂度度量、数据增强缺陷及对抗攻击
许多自然现象具有自相似性，如人脸对称性或歌曲的重复旋律。理解这些模式有助于揭示复杂系统背后的机制。为此，我们提出一种受几何启发的框架，用于在人工神经网络中研究此类现象。我们引入了受19世纪康托集三进制构造启发的CantorNet，该集合是线段上一个自相似且具有反直觉性质的不可数无限零测集。类似地，CantorNet作为一类ReLU神经网络，覆盖了从线性到指数级所有可能的科莫戈罗夫复杂度（以描述长度衡量），其决策边界可任意锯齿状但解析形式已知。该模型不仅可作为复杂度度量的测试平台，还能揭示忽略几何结构的数据增强和对抗攻击中的潜在陷阱。
原文摘要 · Abstract (English)
Many natural phenomena are characterized by self-similarity, for example the symmetry of human faces, or a repetitive motif of a song. Studying of such symmetries will allow us to gain deeper insights into the underlying mechanisms of complex systems. Recognizing the importance of understanding these patterns, we propose a geometrically inspired framework to study such phenomena in artificial neural networks. To this end, we introduce \emph{CantorNet}, inspired by the triadic construction of the Cantor set, which was introduced by Georg Cantor in the $19^\text{th}$ century. In mathematics, the Cantor set is a set of points lying on a single line that is self-similar and has a counter intuitive property of being an uncountably infinite null set. Similarly, we introduce CantorNet as a sandbox for studying self-similarity by means of novel topological and geometrical complexity measures. CantorNet constitutes a family of ReLU neural networks that spans the whole spectrum of possible Kolmogorov complexities, including the two opposite descriptions (linear and exponential as measured by the description length). CantorNet's decision boundaries can be arbitrarily ragged, yet are analytically known. Besides serving as a testing ground for complexity measures, our work may serve to illustrate potential pitfalls in geometry-ignorant data augmentation techniques and adversarial attacks.
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