Recovery of Exact Sparse Representations in the Presence of Bounded Noise

The purpose of this contribution is to extend some recent results on sparse representations of signals in redundant bases developed in the noise-free case to the case of noisy observations. The type of question addressed so far is as follows: given an (n,m)-matrix A with m>n and a vector b=Ax <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> , i.e., admitting a sparse representation x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> , find a sufficient condition for b to have a unique sparsest representation. The answer is a bound on the number of nonzero entries in x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> . We consider the case b=Ax <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> +e where x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> satisfies the sparsity conditions requested in the noise-free case and e is a vector of additive noise or modeling errors, and seek conditions under which x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> can be recovered from b in a sense to be defined. The conditions we obtain relate the noise energy to the signal level as well as to a parameter of the quadratic program we use to recover the unknown sparsest representation. When the signal-to-noise ratio is large enough, all the components of the signal are still present when the noise is deleted; otherwise, the smallest components of the signal are themselves erased in a quite rational and predictable way

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