1 research outputs found
Universally Elevating the Phase Transition Performance of Compressed Sensing: Non-Isometric Matrices are Not Necessarily Bad Matrices
In compressed sensing problems, 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 minimization. \cite{Icassp reweighted l_1}, \cite{Isit
reweighted l_1}, \cite{XuScaingLaw} and \cite{iterativereweightedjournal} have
shown that a two-stage re-weighted minimization algorithm can boost
the phase transition performance for signals whose nonzero elements follow an
amplitude probability density function (pdf) whose -th derivative
for some integer . However, for signals whose
nonzero elements are strictly suspended from zero in distribution (for example,
constant-modulus, only taking values `' or `' for some nonzero real
number ), no polynomial-time signal recovery algorithms were known to
provide better phase transition performance than plain 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 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.Comment: 6pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1010.2236, arXiv:1004.040