arXiv：cs.LG· Kosio Beshkov·· 13 小时前
神经网络拓扑表达力的构造性上界研究
Constructive Bounds for the Topological Expressivity of Neural Networks
AI 导读
研究者提出用神经网络输出 $\beta(\Phi(X))$ 的 Betti 数来研究拓扑表达力，替代仅适用于二分类的决策边界 Betti 数方法。证明了一个 ReLU 网络能以特定架构表达与球面楔和同伦等价的空间，并通过对 $N$ 个球面乘积的构造获得了更优的 Betti 数上界。最后猜想球面同胚空间存在更强的紧致界，对应架构为 $(d,d+1,d+1)$。
正文
Abstract:The expressivity of a neural network $\Phi$ describes the class of functions that a network can express. The term topological expressivity refers to a family of measures particularly suited to understand binary classification and is often formulated in terms of the Betti numbers of a decision boundary $\beta (\Phi^{-1}(0))$. Despite its popularity, this measure is not as useful for regression and other problems that do not involve a decision boundary. Instead, we propose to study topological expressivity in terms of the Betti numbers of the output of a neural network $\beta(\Phi(X))$. We formally prove that a ReLU network with architecture $(d,\sum\limits_{k=1}^{d
} \beta_k(3k+2),\sum\limits_{k=1}^{d}\beta_k(k+1))$ can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gives upper bounds for any arbitrary connected ($\beta_0=1$) sequence of Betti numbers, but by constructing spaces homotopy equivalent to products of $N$ spheres of dimension $d_1,...,d_N$ we achieve much better bounds $(\sum\limits_{n=1} d_n, 3\sum\limits_{n=1}\limits^N d_n + 2N, \sum\limits_{n=1}\limits^N d_n + N) $ for Betti numbers given by $\prod_n (1+x^{d_n})$. Finally we conjecture a much stronger tight bound which gives the architectural bound $(d,d+1,d+1)$ for a space homeomorphic to the sphere.
| Subjects: | Algebraic Topology (math.AT); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.13931 [math.AT] |
| (or arXiv:2610.13931v1 [math.AT] for this version) | |
| https://doi.org/10.48550/arXiv.2610.13931 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Kosio Beshkov [view email]
[v1]
Fri, 9 Oct 2026 17:55:33 UTC (773 KB)
来源：arXiv：cs.LG · arxiv.org