Improved subspace estimation for multivariate observations of high dimension: the deterministic signals case

TL;DR

This paper introduces new consistent subspace estimation methods for high-dimensional multivariate data with deterministic signals, overcoming biases of traditional methods in large-sample asymptotic regimes.

Contribution

It proposes novel subspace estimators applicable to deterministic signals, extending previous random signal models, and provides new eigenvalue location results for sample covariance matrices.

Findings

Proposed estimators outperform standard methods in high-dimensional settings.

New eigenvalue location results for Gaussian models are established.

Method is robust regardless of source signal statistical properties.

Abstract

We consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude. In this context, traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones. It has recently been suggested that this situation (where the sample size is small compared to the observation dimension) can be very accurately modeled by considering the asymptotic regime where the observation dimension $M$ and the number of snapshots $N$ converge to $+\infty$ at the same rate. Using large random matrix theory results, it can be shown that traditional subspace estimates are not consistent in this asymptotic regime. Furthermore, new consistent subspace estimate can be proposed, which outperform the standard subspace methods for realistic values of $M$ and $N$. The work carried out so far in this area has always been based on the assumption that the observations are random, independent and identically distributed in the time domain. The goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals. In practice, this allows to use the proposed approach regardless of the statistical properties of the source signals. In order to construct the proposed estimators, new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of Information plus Noise complex Gaussian models are established. These results are believed to be of independent interest.