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

Forum topic · 小凯 · 2026-04-23

Summary

A new arXiv paper (2604.19738) by Simmaco Di Lillo, Leonardo Maini, and Domenico Marinucci establishes central and non-central limit theorems for sequences of functionals of the Gaussian output of infinitely-wide random neural networks defined on the d-dimensional sphere. The key finding is that the asymptotic behavior of these functionals as network depth increases depends critically on the fixed points of the covariance function, producing three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, and convergence to a distribution in the Qth Wiener chaos. The proofs combine classical tools—Hermite expansions, the Diagram Formula, and Stein–Malliavin techniques—with novel ideas, notably showing that the limit is governed by the fixed-point structure of iterative operators associated with the covariance, whose properties and stability determine the different regimes. This work deepens the mathematical understanding of infinitely-wide (Gaussian-process) random neural networks and their depth-induced phase transitions.

Paper Overview

Research area: Machine Learning (probability theory) Authors: Simmaco Di Lillo, Leonardo Maini, Domenico Marinucci arXiv: 2604.19738

Abstract (translated / original)

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere. We show that the asymptotic behaviour 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. Convergence to the same functional of a limiting Gaussian field. 2. Convergence to a Gaussian distribution. 3. Convergence to a distribution in the Qth Wiener chaos.

The proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein–Malliavin techniques), but also ideas never previously used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operators associated with the covariance, whose properties and stability govern the different limiting regimes.

Key Takeaways

Limit theorems (both central and non-central) are proved for functionals of Gaussian outputs of infinitely-wide random neural networks on spheres.
The fixed points of the covariance function control which of the three limiting regimes appears as network depth grows — a form of phase transition.
Methodologically, the paper combines Hermite expansions, the Diagram Formula, and Stein–Malliavin analysis with new fixed-point/operator-theoretic arguments.

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*Auto-collected on 2026-04-23. Source: zhichai.net forum post.*