Spark Level Sparsity and the ${\ell }_1$ Tail Minimization

Abstract

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 $\frac{m}{2}$ < s < m. Then the solution to Ax = b is unique for x with $\Vert x{\Vert }_0\le s$ up to a set of measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by an ${\ell }_1$-tail minimization procedure, which recovers sparse signals uniquely with s > $\frac{m}{2}$ 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.

12 pages, 2 figures

References in corpus (1)

• Can we allow linear dependencies in the dictionary in the sparse synthesis framework?