Universally Elevating the Phase Transition Performance of Compressed Sensing: Non-Isometric Matrices are Not Necessarily Bad Matrices

In compressed sensing problems, l1 minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomialtime algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than l1 minimization. [20], [21], [22] and [23] have shown that a two-stage re-weighted l1 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 (0) 6= 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 polynomialtime signal recovery algorithms were known to provide better phase transition performance than plain l1 minimization, especially for dense sensing matrices. In this paper, we show that a polynomial-time algorithm can universally elevate the phasetransition performance of compressed sensing, compared with l1 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 nonisometric 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.