Estimating principal components of covariance matrices using the Nystr\"{o}m method

TL;DR

This paper introduces a Nyström-based covariance estimator that efficiently approximates principal components, improves estimation accuracy in high-dimensional settings, and is applicable to tasks like beamforming and image denoising.

Contribution

It presents a novel covariance estimation method using the Nyström approach that is computationally efficient and includes theoretical error analysis and practical applications.

Findings

The estimator reduces computational complexity for high-dimensional covariance matrices.

It improves the accuracy of principal component estimation in small sample scenarios.

Empirical results demonstrate effectiveness in adaptive beamforming and image denoising.

Abstract

Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a low-dimensional principal subspace via their spectral decomposition. However, exact eigenanalysis is computationally intractable for sufficiently high-dimensional matrices, and in the case of small sample sizes, sample eigenvalues and eigenvectors are known to be poor estimators of their population counterparts. To address these issues, we propose a covariance estimator that is computationally efficient while also performing shrinkage on the sample eigenvalues. Our approach is based on the Nystr\"{o}m method, which uses a data-dependent orthogonal projection to obtain a fast low-rank approximation of a large positive semidefinite matrix. We provide a theoretical analysis of the error properties of our estimator as well as empirical results, including examples of its application to adaptive beamforming and image denoising.