# The Spectrum of Random Kernel Matrices

# The Spectrum of Random Kernel Matrices

We consider $n-by-n$ matrices whose $(i, j)-th$ entry is $f(X_i^T X_j)$, where $X_1, ...,X_n$ are i.i.d. standard Gaussian random vectors in $R^p$, and the kenrel function f is a real-valued function. We study the weak limit of the spectral density when p and n go to infinity and $p/n = \gamma$ which is a constant. The limiting spectral density is dictated by a cubic equation involving its Stieltjes transform, and the parameters of the cubic equation are decided by the Hermite expansion of the rescaled kernel function. While the case of kernel functions that are differentiable at the origin has been previously resolved by El-Karoui (2010), our result is applicable to non-smooth kernel functions, e.g. the sign function. For this larger class of kernel functions, we obtain a new family of limiting densities, which includes the Marcenko-Pastur distribution and Wigner’s semi-circle as special cases.