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Abstract: Compressed Sensing decoding algorithms can efficiently recover an Ndimensional real-valued vector x to within a factor of its best k-termapproximation by taking m = 2klogN-k measurements y = Phi x. If the sparsityor approximate sparsity level of x were known, then this theoretical guaranteewould imply quality assurance of the resulting compressed sensing estimate.However, because the underlying sparsity of the signal x is unknown, thequality of a compressed sensing estimate x* using m measurements is notassured. Nevertheless, we demonstrate that sharp bounds on the error || x - x*|| 2 can be achieved with almost no effort. More precisely, we assume that amaximum number of measurements m is pre-imposed; we reserve 4logp of theoriginal m measurements and compute a sequence of possible estimatesx j {j=1}^p to x from the m - 4logp remaining measurements; the errors ||x- x* j || 2 for j = 1,

., p can then be bounded with high probability. As aconsequence, numerical upper and lower bounds on the error between x and thebest k-term approximation to x can be estimated for p values of k with almostno cost. Our observation has applications outside of compressed sensing aswell.

Autor: Rachel Ward


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