570 research outputs found
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 leads to sharp results.
It is shown that for any given constant , in compressed sensing
guarantees the exact recovery of all
sparse signals in the noiseless case through the constrained
minimization, and similarly in affine rank minimization
ensures the exact reconstruction of
all matrices with rank at most in the noiseless case via the constrained
nuclear norm minimization. Moreover, for any ,
is not sufficient to guarantee
the exact recovery of all -sparse signals for large . Similar result also
holds for matrix recovery. In addition, the conditions and are also shown to
be sufficient respectively for stable recovery of approximately sparse signals
and low-rank matrices in the noisy case.Comment: to appear in IEEE Transactions on Information Theor
Polar Polytopes and Recovery of Sparse Representations
Suppose we have a signal y which we wish to represent using a linear
combination of a number of basis atoms a_i, y=sum_i x_i a_i = Ax. The problem
of finding the minimum L0 norm representation for y is a hard problem. The
Basis Pursuit (BP) approach proposes to find the minimum L1 norm representation
instead, which corresponds to a linear program (LP) that can be solved using
modern LP techniques, and several recent authors have given conditions for the
BP (minimum L1 norm) and sparse (minimum L0 solutions) representations to be
identical. In this paper, we explore this sparse representation problem} using
the geometry of convex polytopes, as recently introduced into the field by
Donoho. By considering the dual LP we find that the so-called polar polytope P
of the centrally-symmetric polytope P whose vertices are the atom pairs +-a_i
is particularly helpful in providing us with geometrical insight into
optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In
exploring this geometry we are able to tighten some of these earlier results,
showing for example that the Fuchs condition is both necessary and sufficient
for L1-unique-optimality, and that there are situations where Orthogonal
Matching Pursuit (OMP) can eventually find all L1-unique-optimal solutions with
m nonzeros even if ERC fails for m, if allowed to run for more than m steps
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