Spectral Analysis of Nonlinear Operators: Theory and Applications to Neural Networks and Optimization

Abstract

This paper presents a nonlinear spectral framework for analyzing monotone and nonexpansive operators in Banach and Hilbert spaces. We construct a nonlinear spectral resolution for maximal monotone operators using Yosida approximations and Fitzpatrick functions, leading to a family of nonlinear projections and an associated spectral measure. For nonexpansive mappings, we establish an iterative spectral approximation based on Krasnoselskii iterations, with proven convergence and recovery of nonlinear eigenvectors. We further extend this framework to ReLU-based neural networks, analyzing spectral bounds, depth-dependent scaling, and gradient alignment. These results bridge nonlinear operator theory and neural architectures, offering new tools for theoretical analysis and applications in optimization, physics, and machine learning.