Fixed Point Continuation (FPC). General l1-regularized minimization problems of the form (1) min ||x||1 + μ f(x), where f is a convex, but not necessarily strictly convex, function, can be solved with a globally-convergent fixed-point iteration scheme. In addition, q-linear rates of convergence can be achieved under mild conditions. Problems in the form of (1) are often of interest when x is expected to be sparse, or contain outliers. In compressed sensing signal reconstruction, f(x) is a weighted least-squares term. In this case, q-linear convergence rates can be shown as long as a certain reduced Hessian is full rank, or a strict complementarity condition holds. In order to obtain good practical performance, the basic fixed-point iterations should be augmented with a continuation approach. In brief, the continuation approach consists of solving (1) for an increasing sequence of μ values, using the solution at the last μ value as the starting point for the next μ value. Thus, Fixed-Point Continuation (FPC).

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