mlsanityData to 14 Oct 2026

Constructive Bounds for the Topological Expressivity of Neural Networks

arXiv 2610.13931 ↗First posted 9 Oct 2026math.AT cs.LGcited by 0

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Abstract (arXiv)

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 β (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 β(Phi(X)). We formally prove that a ReLU network with architecture (d,sumlimitsk=1d βk(3k+2),sumlimitsk=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 (sumlimitsn=1 dn, 3sumlimitsn=1limitsN dn + 2N, sumlimitsn=1limitsN dn + N) for Betti numbers given by prodn (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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References · 23

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11 more
  1. 2020Topology of deep neural networkscited by 112
  2. 2018Tropical Geometry of Deep Neural Networkscited by 95
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