Spectral Phase Transitions in Non-Linear Wigner Spiked Models

2023/10/21 by Alice Guionnet, Justin Ko, Guionnet, Alice +7 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #60B20 #Blind Source Separation Techniques #FOS: Mathematics #Probability (math.PR) #Quantum optics and atomic interactions #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2310.14055

openalex publication_date 2023/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic behavior of the spectrum of a random matrix where a non-linearity is applied entry-wise to a Wigner matrix perturbed by a rank-one spike with independent and identically distributed entries. In this setting, we show that when the signal-to-noise ratio scale as N^{(1)/(2) (1-1/k_⋆)}, where k_⋆ is the first non-zero generalized information coefficient of the function, the non-linear spike model effectively behaves as an equivalent spiked Wigner matrix, where the former spike before the non-linearity is now raised to a power k_⋆. This allows us to study the phase transition of the leading eigenvalues, generalizing part of the work of Baik, Ben Arous and Peché to these non-linear models.

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• Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares
• Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling

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