The Spectrum of Random Inner-product Kernel Matrices

Understand

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 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 and n go to infinity, p/n = \gamma which is a constant, and f is properly scaled so that Var(f(X_i^T X_j)) is O(p^{-1}), the spectral density converges weakly to a limiting density on R.