Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity

TL;DR

This paper analyzes the complexity of high-dimensional random landscapes formed by superimposing many random plane waves, using a novel matrix approach related to the Gaussian Marchenko-Pastur ensemble.

Contribution

It introduces a spectral matrix framework to compute the landscape complexity and spectral properties of superimposed random plane waves in high dimensions.

Findings

Derived the mean spectral density of the associated matrices.

Computed moments and correlation functions of characteristic polynomial products.

Established a connection to the Gaussian Marchenko-Pastur ensemble.

Abstract

Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in $\mathbb{R}^N$ obtained by superimposing $M>N$ plane waves of random wavevectors and amplitudes. For this landscape we show how to compute the "annealed complexity" controlling the asymptotic growth rate of the mean number of stationary points as $N\to \infty$ at fixed ratio $\alpha=M/N>1$. The framework of this computation requires us to study spectral properties of $N\times N$ matrices $W=KTK^T$, where $T$ is diagonal with $M$ mean zero 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 and related matrices.