Data-Driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application

TL;DR

This paper introduces a data-driven optimal shrinkage algorithm, eOptShrink, for matrix denoising in high-dimensional settings with dependent, colored noise, providing theoretical guarantees and practical applications in ECG signal extraction.

Contribution

The paper develops the first data-driven optimal shrinkage method for matrices with separable covariance noise, without needing to estimate the noise covariance structure.

Findings

eOptShrink outperforms existing algorithms in numerical simulations.

Theoretical convergence guarantees are established for the estimators.

Successful application to fetal ECG extraction demonstrates practical utility.

Abstract

We develop a data-driven optimal shrinkage algorithm for matrix denoising in the presence of high-dimensional noise with a separable covariance structure; that is, the noise is colored and dependent across samples. The algorithm, coined {\em extended OptShrink} (eOptShrink) depends on the asymptotic behavior of singular values and singular vectors of the random matrix associated with the noisy data. Based on the developed theory, including the sticking property of non-outlier singular values and delocalization of the non-outlier singular vectors associated with weak signals with a convergence rate, and the spectral behavior of outlier singular values and vectors, we develop three estimators, each of these has its own interest. First, we design a novel rank estimator, based on which we provide an estimator for the spectral distribution of the pure noise matrix, and hence the optimal shrinker called eOptShrink. In this algorithm we do not need to estimate the separable covariance structure of the noise. A theoretical guarantee of these estimators with a convergence rate is given. On the application side, in addition to a series of numerical simulations with a comparison with various state-of-the-art optimal shrinkage algorithms, we apply eOptShrink to extract maternal and fetal electrocardiograms from the single channel trans-abdominal maternal electrocardiogram.