Compressed Sensing via Iterative Support Detection

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Abstract
We present a new compressive sensing reconstruction method "ISD". ISD addresses failed cases of L1-based construction due to insufficient measurements. ISD will learn from wrong solutions and come up with new minimization problems that return signals that are either correct or better. Specifically, from an incorrect signal ISD detects an index set I that includes components most likely to be true nonzeros, obtains a new signal x by solving min{sum_{i not in I} |x_i| : Ax = b}, and repeats such support detection and minimization using latest x and I from one another until convergence. We introduce an efficient implementation of ISD, called threshold-ISD, for recovering signals with fast decaying distributions of nonzeros from compressive measurements. Numerical experiments show that threshold-ISD has significant overall advantages over the classical L1 minimization approach, as well as two other state-of-the-art algorithms such as the iterative reweighted L1 minimization algorithm (IRL1) and the iterative reweighted least-squares algorithm (IRLS).

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