Masters Thesis

The tail minimization algorithm and compressed sensing

₀ ≤ s up to a set of measure 0 in every s-sparse plane. An iterative l₁-tail minimization procedure is studied which recovers sparse signals uniquely with s spark(A) - 1. A subset nullspace property is found, and is necessary and sufficient for tail minimization to succeed. An error bound formula for signal reconstruction using a tail minimization procedure is also obtained. We further show instead that the mere l₁-minimization would actually fail if s spark(A)-1 /2 from the same measure theoretical point of view. A subset dictionary nullspace property (subset D-NSP) is derived as a necessary and sufficient condition for tail minimization with coherent frames.xIt 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. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements Ax = b. Specifically, suppose spark(A)-1/ 2 s spark(A) - 1. Then the solution to Ax = b is unique for x with

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