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

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.