Sparsity-based Cholesky Factorization and its Application to Hyperspectral Anomaly Detection

TL;DR

This paper introduces two sparsity-based methods for estimating large covariance matrices via Cholesky factorization, improving hyperspectral anomaly detection accuracy.

Contribution

It proposes novel sparsity-inducing techniques for Cholesky factor estimation that ensure positive definiteness and enhance anomaly detection performance.

Findings

Effective covariance estimators demonstrated through simulations

Improved hyperspectral anomaly detection results

Guarantees positive definiteness of estimated matrices

Abstract

Estimating large covariance matrices has been a longstanding important problem in many applications and has attracted increased attention over several decades. This paper deals with two methods based on pre-existing works to impose sparsity on the covariance matrix via its unit lower triangular matrix (aka Cholesky factor) $\mathbf{T}$. The first method serves to estimate the entries of $\mathbf{T}$ using the Ordinary Least Squares (OLS), then imposes sparsity by exploiting some generalized thresholding techniques such as Soft and Smoothly Clipped Absolute Deviation (SCAD). The second method directly estimates a sparse version of $\mathbf{T}$ by penalizing the negative normal log-likelihood with $L_1$ and SCAD penalty functions. The resulting covariance estimators are always guaranteed to be positive definite. Some Monte-Carlo simulations as well as experimental data demonstrate the effectiveness of our estimators for hyperspectral anomaly detection using the Kelly anomaly detector.