We study the asymptotic spectral distribution 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 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 $α\in ]0,2[$ and sparse matrices with $\mathcal{O}(1)$ nonzero entries per row. The activation function $f$, applied entrywise, is bounded, smooth, odd, and nonlinear. We compute the limiting eigenvalue distribution of $YY^\top$ 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.

翻译：我们研究了两层神经网络模型产生的共轭核随机矩阵 $YY^\top$ 的渐近谱分布，其中 $Y= f(WX)$。我们考虑 $W$ 和 $X$ 为具有独立同分布元素的随机矩形矩阵的情形：$W$ 的元素服从重尾分布，而 $X$ 的元素具有轻尾。我们对 $W$ 的假设涵盖了一类广泛的重尾分布，例如指数 $α\in ]0,2[$ 的对称 $α$-稳定律，以及每行具有 $\mathcal{O}(1)$ 个非零元素的稀疏矩阵。按元素施加的激活函数 $f$ 是有界、光滑、奇性且非线性的。我们通过矩计算 $YY^\top$ 的极限特征值分布，并证明重尾权重会导致 $Y$ 的元素之间产生强相关性，从而产生比轻尾情形更丰富且本质上不同的谱行为。