Global law of conjugate kernel random matrices with heavy-tailed weights

Abstract: We study the asymptotic spectral behavior of the conjugate kernel random matrix $YY^\top$, where $Y= f(WX)$ arises from a two-layer neural network model. We consider the setting where $W$ and $X$ are both 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 $\alpha$-stable laws with $\alpha \in (0,2)$ and sparse matrices with $\mathcal{O}(1)$ nonzero entries per row. The activation function $f$, applied entrywise, is nonlinear, smooth, and odd. By computing the eigenvalue distribution of $YY^\top$ through its moments, we show that heavy-tailed weights induce strong correlations between the entries of $Y$, leading to richer and fundamentally different spectral behavior compared to models with light-tailed weights.