On the Topology of Neural Network Superlevel Sets
Machine Learning
2026-03-04 v1 Optimization and Control
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.
Cite
@article{arxiv.2603.02973,
title = {On the Topology of Neural Network Superlevel Sets},
author = {Bahman Gharesifard},
journal= {arXiv preprint arXiv:2603.02973},
year = {2026}
}