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A novel methodology is employed to develop algorithms for computing sparse solutions to linear inverse problems, starting from suitably defined diversity measures whose minimization promotes sparsity. These measures include p-norm-like (/spl Lscr//sub (p/spl les/1)/) diversity measures, and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation of the gradient, and involves successive relaxation of the Lagrangian necessary condition. The general nature of the methodology provides a systematic approach for deriving a class of algorithms called FOCUSS (FOCal Underdetermined System Solver), and a natural mechanism for extending them.