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.

Editor's note: Rostam J. Neuwirth is a professor of the Faculty of Law, University of Macau. The article reflects the author's opinions, and not necessarily the views of CGTN. At the end of last year, the General Conference of the United Nations Educational, Scientific and Cultural Organization adopted the Recommendation on the Ethics of Artificial Intelligence as the first global standard-setting instrument responding to the ethical concerns related to artificial intelligence (AI). This recommendation must strongly be welcome, because – albeit legally non-binding – it reflects an emerging global consensus on the ethical concerns raised and serious risks caused by AI. Most of all, it recognizes the need to work on global solutions to this problem of fundamental importance for present and future generations.