Constructive Bounds for the Topological Expressivity of Neural Networks
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))...
| Authors | Kosio Beshkov |
|---|---|
| Primary category | math.AT |
| Published | 9 October 2026 |
| Last updated at source | 9 October 2026 |
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