On the Topology of Neural Network Superlevel Sets

Bahman Gharesifard

cs.LGMar 3, 2026

On the Topology of Neural Network Superlevel Sets

Authors: Bahman Gharesifard

Abstract

We show that neural networks with activations satisfying a Riccati-type ordinary differential equation condition, an assumption arising in recent universal approximation results in the uniform topology, produce Pfaffian outputs on analytic domains with format controlled only by the architecture. Consequently, superlevel sets, as well as Lie bracket rank drop loci for neural network parameterized vector fields, admit architecture-only bounds on topological complexity, in particular on total Betti numbers, uniformly over all weights.

Figures & tables

Panel of Figure 1 — Figure 1. A scalar map $F:V\to\mathbb{R}$ and its decision region $D=\{F\geq 0\}=F^{-1}([0,\infty))$ are depicted schematically. The dashed curves represent the zero level sets.

Panel of Figure 2 — Figure 2. Assuming analyticity, each change of sign can only occur at a zero of $F$ , so the number of interval components is controlled by $\Zeros(F;I)$ .

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Constructive Bounds for the Topological Expressivity of Neural Networks

Kosio Beshkov

Department of Physics, University of Oslo, Oslo, Norway.

Neural Network Expressivity ReLU Neural Networks

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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 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

\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.

Abstract

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