[论文] Constructive Bounds for the Topological Expressivity of Neural Network...

论文概要

研究领域: ML 作者: Kosio Beshkov 发布时间: 2026-10-09 arXiv: 2610.13931

中文摘要

神经网络的表达性描述了网络可表达的函数类，拓扑表达性是一组尤其适合理解二分类的度量，通常以决策边界的 Betti 数 β(Φ⁻¹(0)) 来表述。尽管流行，该度量对回归及不涉及决策边界的其他问题用处有限。本文改为研究网络输出的 Betti 数 β(Φ(X)) 所刻画的拓扑表达性。我们形式化地证明：具有 (d, Σβₖ(3k+2), Σβₖ(k+1)) 结构的 ReLU 网络可以表达同伦等价于任意 Betti 数的球楔和的空间。这为任意连通（β₀=1）Betti 数序列给出了上界；进一步通过构造同伦等价于 N 个球面乘积的空间，对 Betti 数为 Π(1+x^{dₙ}) 的情形得到更好的界 (Σdₙ, 3Σdₙ+2N, Σdₙ+N)。最后我们猜想一个更强的紧界，对与球面同胚的空间给出 (d, d+1, d+1) 的结构上界。

原文摘要

The expressivity of a neural network \(Φ\) 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 \(β(Φ^{-1}(0))\). 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 \(β(Φ(X))\). We formally prove that a ReLU network with architecture \((d,\sum\limits_{k=1}^{d } β_k(3k+2),\sum\limits_{k=1}^{d}β_k(k+1))\) can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gi...

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