Joint Sparse Recovery for Signals of Spark-Level Sparsity and MMV Tail-$\ell _{2,1}$ Minimization
Baifu Zheng, Cao Zeng, Shidong Li, Guisheng Liao
Abstract
The rank of the sparse signals brought by multiple measurement vectors (MMV) augments the performance of joint sparse recovery. In general, suppose the sparsity level k is less than or equal to [rank( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> )+spark( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> )-1]/2, the sparsest solution of the MMV problem is unique and recoverable via various methods. It is shown in this letter that the unique solution of the sparsity level k up to spark( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> )-1 actually exists in a measure theoretical point of view. More specifically, even when [rank( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> )+spark( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> )-1]/2 ≤ k <; spark( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> ), the sparsest solution to <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> = <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</b> is still unique with full Lebesgue measure in every k-sparse coordinate space. This phenomenon is fully confirmed by the MMV tail- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> minimization technique. Furthermore, the phenomenon that the traditional <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> minimization actually fails to recover <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> with k ≥ [spark( <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> )-1]/2 is investigated from the same perspective of measure theory. Extensive numerical tests conducted by the MMV tail- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> minimization and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> minimization are demonstrated to confirm the findings. The tail minimization procedure exhibits the most prominent effectiveness for the larger sparsity levels among all known techniques.