Performance Bounds for Sparse Parametric Covariance Estimation in Gaussian Models

TL;DR

This paper derives theoretical lower bounds on the variance of estimators for sparse covariance matrices in Gaussian models, using RKHS theory, and compares these bounds with standard estimators.

Contribution

It introduces a novel application of RKHS theory to establish variance bounds for sparse covariance estimation in Gaussian models.

Findings

Lower bounds on estimator variance derived using RKHS theory

Comparison shows how standard estimators perform relative to theoretical bounds

Provides insights into the efficiency of sparse covariance estimators

Abstract

We consider estimation of a sparse parameter vector that determines the covariance matrix of a Gaussian random vector via a sparse expansion into known "basis matrices". Using the theory of reproducing kernel Hilbert spaces, we derive lower bounds on the variance of estimators with a given mean function. This includes unbiased estimation as a special case. We also present a numerical comparison of our lower bounds with the variance of two standard estimators (hard-thresholding estimator and maximum likelihood estimator).