Constructive Bounds for the Topological Expressivity of Neural Networks
Abstract
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,∑k=1d βk(3k+2),∑k=1dβk(k+1)) can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gives upper bounds for any arbitrary connected (β0=1) sequence of Betti numbers, but by constructing spaces homotopy equivalent to products of N spheres of dimension d1,...,dN 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 ∏n (1+xdn). 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.
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BibTeX
@misc{beshkov2026constructive,
title = {{Constructive Bounds for the Topological Expressivity of Neural Networks}},
author = {Kosio Beshkov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13931},
eprint = {2610.13931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13931},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Constructive Bounds for the Topological Expressivity of Neural Networks
AU - Kosio Beshkov
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13931
UR - https://arxiv.org/abs/2610.13931
AB - 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,∑_k=1^d β_k(3k+2),∑_k=1^dβ_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 (β_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 ∏_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.
ER -