Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices

This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool, which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while yielding sharp results. It is shown that for any given constant
$t\geq{4/3}$
, in compressed sensing,
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$
guarantees the exact recovery of all
$k$
sparse signals in the noiseless case through the constrained
$\ell_{1}$
minimization, and similarly, in affine rank minimization,
$\delta_{tr}^{\cal M}<\sqrt{(t-1)/t}$
ensures the exact reconstruction of all matrices with rank at most
$r$
in the noiseless case via the constrained nuclear norm minimization. In addition, for any
$\epsilon>0$
,
$\delta_{tk}^{A}<\sqrt{{t-1}\over{t}}+\epsilon$
is not sufficient to guarantee the exact recovery of all
$k$
-sparse signals for large
$k$
. Similar results also hold for matrix recovery. In addition, the conditions
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$
and
$\delta_{tr}^{\cal M}<\sqrt{(t-1)/t}$
are also shown to be sufficient, respectively, for stable recovery of approximately sparse signals and low-rank matrices in the noisy case.