2,692 research outputs found
Relaxed Recovery Conditions for OMP/OLS by Exploiting both Coherence and Decay
We propose extended coherence-based conditions for exact sparse support
recovery using orthogonal matching pursuit (OMP) and orthogonal least squares
(OLS). Unlike standard uniform guarantees, we embed some information about the
decay of the sparse vector coefficients in our conditions. As a result, the
standard condition (where denotes the mutual coherence and
the sparsity level) can be weakened as soon as the non-zero coefficients
obey some decay, both in the noiseless and the bounded-noise scenarios.
Furthermore, the resulting condition is approaching for strongly
decaying sparse signals. Finally, in the noiseless setting, we prove that the
proposed conditions, in particular the bound , are the tightest
achievable guarantees based on mutual coherence
Exact Recovery Conditions for Sparse Representations with Partial Support Information
We address the exact recovery of a k-sparse vector in the noiseless setting
when some partial information on the support is available. This partial
information takes the form of either a subset of the true support or an
approximate subset including wrong atoms as well. We derive a new sufficient
and worst-case necessary (in some sense) condition for the success of some
procedures based on lp-relaxation, Orthogonal Matching Pursuit (OMP) and
Orthogonal Least Squares (OLS). Our result is based on the coherence "mu" of
the dictionary and relaxes the well-known condition mu<1/(2k-1) ensuring the
recovery of any k-sparse vector in the non-informed setup. It reads
mu<1/(2k-g+b-1) when the informed support is composed of g good atoms and b
wrong atoms. We emphasize that our condition is complementary to some
restricted-isometry based conditions by showing that none of them implies the
other.
Because this mutual coherence condition is common to all procedures, we carry
out a finer analysis based on the Null Space Property (NSP) and the Exact
Recovery Condition (ERC). Connections are established regarding the
characterization of lp-relaxation procedures and OMP in the informed setup.
First, we emphasize that the truncated NSP enjoys an ordering property when p
is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the
truncated NSP for the informed l1 problem, and the truncated NSP for p<1.Comment: arXiv admin note: substantial text overlap with arXiv:1211.728
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