Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spheres

The article titled "The stability of shallow neural networks on spheres: A sharp spectral analysis," published on November 5, 2025, examines the stability properties of shallow neural networks defined on spherical domains. It specifically focuses on analyzing the condition numbers of mass and stiffness matrices associated with these networks. A key contribution of the work is the derivation of sharp asymptotic estimates for the eigenvalues of these matrices, which are crucial for understanding network stability. These estimates are obtained under the assumption that the network's parameters are arranged in an antipodally quasi-uniform manner on the sphere. This spectral analysis provides precise insights into how the geometric distribution of parameters influences the numerical behavior of shallow spherical neural networks. The study contributes to the broader field of neural network theory by addressing stability issues through rigorous mathematical tools. It also aligns with recent research trends on neural networks and their spectral properties, as indicated by related works in the arXiv repository.