Joint Sparse Recovery for Signals of Spark-Level Sparsity and MMV Tail-$\ell _{2,1}$ Minimization

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(
X
)+spark(
A
)-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(
A
)-1 actually exists in a measure theoretical point of view. More specifically, even when [rank(
X
)+spark(
A
)-1]/2 ≤ k <; spark(
A
), the sparsest solution to
A
X
=
Y
is still unique with full Lebesgue measure in every k-sparse coordinate space. This phenomenon is fully confirmed by the MMV tail-
l
_2,1
minimization technique. Furthermore, the phenomenon that the traditional
l
_2,1
minimization actually fails to recover
X
with k ≥ [spark(
A
)-1]/2 is investigated from the same perspective of measure theory. Extensive numerical tests conducted by the MMV tail-
l
_2,1
minimization and
l
_2,1
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.