A CLT for an improved subspace estimator with observations of increasing dimensions

TL;DR

This paper proves a central limit theorem for a new subspace estimator in high-dimensional, small-sample scenarios, providing a theoretical foundation and practical error approximation for improved performance analysis.

Contribution

It introduces a CLT for a recent high-dimensional subspace estimator, enhancing understanding of its statistical properties in small sample, large dimension regimes.

Findings

Proves a CLT for the estimator in high-dimensional asymptotics

Provides an accurate numerical approximation of the mean square error

Demonstrates improved performance over traditional estimators in small sample regimes

Abstract

This paper deals with subspace estimation in the small sample size regime, where the number of samples is comparable in magnitude with the observation dimension. The traditional estimators, mostly based on the sample correlation matrix, are known to perform well as long as the number of available samples is much larger than the observation dimension. However, in the small sample size regime, the performance degrades. Recently, based on random matrix theory results, a new subspace estimator was introduced, which was shown to be consistent in the asymptotic regime where the number of samples and the observation dimension converge to infinity at the same rate. In practice, this estimator outperforms the traditional ones even for certain scenarios where the observation dimension is small and of the same order of magnitude as the number of samples. In this paper, we address a performance analysis of this recent estimator, by proving a central limit theorem in the above asymptotic regime. We propose an accurate approximation of the mean square error, which can be evaluated numerically.