SR-TL1: A Square-Root TL1-Norm Framework for Robust SMV DoA Estimation under Highly-Coherent Dictionaries

Abstract

This paper proposes a Square-Root Transformed $L_1$-norm ($SR\text{-}TL_{1}$) sparse recovery framework for single-measurement-vector (SMV) direction of arrival (DoA) estimation under highly-coherent overcomplete dictionaries with angular-dependent array imperfections. The proposed framework combines the square-root Least Absolute Shrinkage and Selection Operator (square-root LASSO) framework which is known to be robust against noise variance with the Transformed $L_1$-norm ($TL_1$-norm), a non-convex penalty that shows a stronger recovery performance than the classical convex $L_1$-norm under highly-coherent dictionaries. We use the Difference of Convex Algorithm (DCA) along with the Alternating Direction Method of Multipliers (ADMM) algorithm to obtain simple, closed-form update rules and provide an efficient implementation that leverages the low-rank nature of the Gram matrix of the dictionary so as to obtain a computational complexity of at most $\mathcal{O}(MN)$ per iteration where $M$ is the array size and $N$ is the length of the dictionary. We additionally provide a convergence guarantee for the DCA iterates of $SR\text{-}TL_{1}$. Experimental verification of the proposed framework shows a lower sensitivity of the regularization hyperparameter to the noise variance level and a competitive recovery performance compared to state-of-the-art baselines.