Global law of conjugate kernel random matrices with heavy-tailed weights
Year: 2026
Type: article
Abstract: We study the asymptotic spectral distribution of the conjugate kernel random matrix YY⊤, where Y=f(WX) arises from a two-layer neural network model. We consider the setting where W and X are random rectangular matrices with i.i.d. entries, where the entries of W follow a heavy-tailed distribution, while those of X have light tails. Our assumptions on W include a broad class of heavy-tailed distributions, such as symmetric α-stable laws with α∈]0,2[ and sparse matrices with O(1) nonzero entries per row. The activation function f, applied entrywise, is bounded, smooth, odd, and nonlinear. We compute the limiting eigenvalue distribution of YY⊤ through its moments and show that heavy-tailed weights induce strong correlations between the entries of Y, resulting in richer and fundamentally different spectral behavior compared to the light-tailed case.
DOI: 10.1214/25-ejp1464
Authors: Alice Guionnet, Vanessa Piccolo
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W7118609330