Spark Level Sparsity and the ℓ1 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 is recoverable by l 1 -minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements A x = b . Specifically, suppose A is of full spark with m rows, and suppose m 2 s m . Then the solution to A x = 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 l 1 -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 l 1 -minimization would actually fail if s > m 2 even from the same measure theoretical point of view.

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