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 $\ell_1$-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 $\frac{m}{2}$ $\frac{m}{2}$ in thousands and thousands of random tests. We further show instead that the mere $\ell_1$-minimization would actually fail if s > $\frac{m}{2}$ even from the same measure theoretical point of view.