Sparse representations: Recovery conditions and fast algorithm for a new criterion

Most applications of sparse representations are based on a combined ℓ<sub>2</sub>-ℓ<sub>1</sub> criterion, where the least-squares-part ensures closeness to the observations and the ℓ<sub>1</sub>-part sparsity. This choice leads to quite efficient algorithms and has a clear connection to maximum likelihood approaches in case of additive Gaussian noise. We replace the least-squares-part by a ℓ<sub>1</sub>-part and investigate the recovery conditions of the so-obtained ℓ<sub>1</sub> - ℓ<sub>1</sub> criterion. We then propose an algorithm, that minimizes the criterion, in a finite number of steps.

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