Universally Elevating the Phase Transition Performance of Compressed Sensing: Non-Isometric Matrices are Not Necessarily Bad Matrices
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In compressed sensing problems, ℓ_1 minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than ℓ_1 minimization. Icassp reweighted l_1, Isit reweighted l_1, XuScaingLaw and iterativereweightedjournal have shown that a two-stage re-weighted ℓ_1 minimization algorithm can boost the phase transition performance for signals whose nonzero elements follow an amplitude probability density function (pdf) f(·) whose t-th derivative f^t(0) ≠ 0 for some integer t ≥ 0. However, for signals whose nonzero elements are strictly suspended from zero in distribution (for example, constant-modulus, only taking values `+d' or `-d' for some nonzero real number d), no polynomial-time signal recovery algorithms were known to provide better phase transition performance than plain ℓ_1 minimization, especially for dense sensing matrices. In this paper, we show that a polynomial-time algorithm can universally elevate the phase-transition performance of compressed sensing, compared with ℓ_1 minimization, even for signals with constant-modulus nonzero elements. Contrary to conventional wisdoms that compressed sensing matrices are desired to be isometric, we show that non-isometric matrices are not necessarily bad sensing matrices. In this paper, we also provide a framework for recovering sparse signals when sensing matrices are not isometric.
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