Phase Transitions in the Fluctuations of Functionals of Random Neural Networks on the Sphere

Paper Overview

• Field: Machine Learning / Probability
• Authors: Simmaco Di Lillo, Leonardo Maini, Domenico Marinucci
• Published: 2026-04-21
• arXiv: 2604.19738

Summary

The authors establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network defined on the d-dimensional sphere. They show that the asymptotic behavior of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes:

1. Regime I: convergence to the same functional of a limiting Gaussian field. 2. Regime II: convergence to a Gaussian distribution. 3. Regime III: convergence to a distribution in the Qth Wiener chaos.

The proofs exploit now-classical tools (Hermite expansions, the Diagram Formula, Stein–Malliavin techniques), but also introduce ideas that have never appeared in similar contexts: in particular, the asymptotic behavior is determined by the structure and stability of fixed points of an iteration operator associated with the covariance function.

Key Contributions

• A unified limit-theoretic framework for functionals of deep, infinitely-wide random neural networks on the sphere.
• Identification of covariance fixed points as the mechanism producing phase transitions between the three limiting regimes.
• Novel use of iteration-operator fixed-point analysis alongside classical Malliavin-calculus methods.

*Automatically collected on 2026-04-23*