Superposition of random plane waves in high spatial dimensions: Random matrix approach to landscape complexity

Abstract

Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in RN obtained by superimposing M > N plane waves of random wavevectors and amplitudes and further restricted by a uniform parabolic confinement in all directions. For this landscape, we show how to compute the “annealed complexity,” controlling the asymptotic growth rate of the mean number of stationary points as N → ∞ at fixed ratio α = M/N > 1. The framework of this computation requires us to study spectral properties of N × N matrices W = KTKT, where T is a diagonal matrix with M mean zero independent and identically distributed (i.i.d.) real normally distributed entries, and all MN entries of K are also i.i.d. real normal random variables. We suggest to call the latter Gaussian Marchenko–Pastur ensemble as such matrices appeared in the seminal 1967 paper by those authors. We compute the associated mean spectral density and evaluate some moments and correlation functions involving products of characteristic polynomials for such matrices.