Fast implementation of a ℓ1 - ℓ1 regularized sparse representations algorithm.

When seeking a sparse representation of a signal on a redundant basis, one replaces generally the quest for the true sparsest model by an ℓ<inf>1</inf> minimization and solves thus a linear program. In the presence of noise one further replaces the exact reconstruction constraint by an approximate one. The ℓ<inf>2</inf>-norm is generally chosen to measure the reconstruction error because of its link with Gaussian noise and the stability and simplicity of the ensuing algorithms, but the ℓ<inf>1</inf>-norm may be preferred in some cases when the noise has heavier tails or in the presence of outliers. We propose to replace the usual ℓ<inf>2</inf> - ℓ<inf>1</inf> regularized criterion by a ℓ<inf>1</inf> - ℓ<inf>1</inf> regularized criterion and show how to construct a fast dedicated optimization algorithm that solves this criterion in a finite number of steps. Since quite often even these fast optimal programs are considered to be too time consuming, we further develop an ad hoc sub-optimal algorithm that could be called the ℓ<inf>1</inf>-matching pursuit algorithm.