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
$\ell _{p}$
-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 noninformed 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
$\ell _{p}$
-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
$\ell _{1}$
problem, and the truncated NSP for
$p< 1$
.