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Abstract: In this paper we propose a variation of the soft-thresholding algorithm for finding sparse approximate solutions of the equation , where as the sparsity of the iterate increases the penalty function changes. In this approach, sufficiently large entries in a sparse iterate are left untouched. The advantage of this approach is that a higher regularization constant can be used, leading to a significant reduction of the total number of iterations. Numerical experiments for sparse recovery problems, also with noisy data, are included. [Copyright &y& Elsevier]