Direction-of-Arrival Estimation Through Exact Continuous ℓ2,0-Norm Relaxation

• Published in: IEEE signal processing letters Vol. 28; pp. 16 - 20
• Main Authors: Soubies, Emmanuel, Chinatto, Adilson, Larzabal, Pascal, Romano, Joao M. T., Blanc-Feraud, Laure
• Format: Journal Article
• Language: English
• Published: New York IEEE 2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 0}</tex-math> </inline-formula>-norm minimization; <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> ell _{2; Algorithms; Antenna arrays; Computational geometry; Convexity; Direction of arrival; Direction-of-arrival estimation; DOA; Estimation; exact relaxations; Minimax technique; Minimization; MMV-sparse optimization; Optimization; Signal processing algorithms; Sparse matrices
• ISSN: 1070-9908; 1558-2361
• DOI: 10.1109/LSP.2020.3042771

On-grid based direction-of-arrival (DOA) estimation methods rely on the resolution of a difficult group-sparse optimization problem that involves the <inline-formula><tex-math notation="LaTeX">\ell _{2,0}</tex-math></inline-formula> pseudo-norm. In this work, we show that an exact relaxation of this problem can be obtained by replacing the <inline-formula><tex-math notation="LaTeX">\ell _{2,0}</tex-math></inline-formula> term with a group minimax concave penalty with suitable parameters. This relaxation is more amenable to non-convex optimization algorithms as it is continuous and admits less local (not global) minimizers than the initial <inline-formula><tex-math notation="LaTeX">\ell _{2,0}</tex-math></inline-formula>-regularized criteria. We then show on numerical simulations that the minimization of the proposed relaxation with an iteratively reweighted <inline-formula><tex-math notation="LaTeX">\ell _{2,1}</tex-math></inline-formula> algorithm leads to an improved performance over traditional approaches.