Spark Level Sparsity and the $\ell_1$ Tail Minimization

Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A, and then the unique sparsest solution exists, and recoverable by l1-minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements Ax = b. Specifically, suppose A is of full spark with m rows, and suppose m 2 < s < m. Then the solution to Ax = b is unique for x with ‖x‖0 ≤ s up to a set of measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by an l1-tail minimization procedure, which recovers sparse signals uniquely with s > m 2 in thousands and thousands of random tests. We further show instead that the mere l1-minimization would actually fail if s > m 2 even from the same measure theoretical point of view.