The Spectrum of Random Inner-product Kernel Matrices

Abstract: We consider n-by-n matrices whose (i, j)-th entry is f(XT i Xj), where X1, . . . ,Xn are i.i.d. standard Gaussian random vectors in Rp, and f is a real-valued function. The eigenvalue distribution of these random kernel matrices is studied at the “large p, large n” regime. It is shown that, when p, n → ∞ and p/n = γ which is a constant, and f is properly scaled so that V ar(f(XT i Xj)) is O(p ), the spectral density converges weakly to a limiting density on R. The limiting density is dictated by a cubic equation involving its Stieltjes transform. While for smooth kernel functions the limiting spectral density has been previously shown to be the Marcenko-Pastur distribution, our analysis is applicable to non-smooth kernel functions, resulting in a new family of limiting densities.