Perturbation Analysis of Orthogonal Matching Pursuit

Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have considered the recovery of a sparse signal
${\mbi {x}}$
through
${\mmb {\Phi}}$
and
${\mbi {y}}={\mmb {\Phi}} {\mbi {x}}+{\mbi {b}}$
, where
${\mmb {\Phi}}$
is a matrix with more columns than rows and
${\mbi {b}}$
denotes the measurement noise. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both
${\mbi {y}}$
and
${\mmb {\Phi}}$
are perturbed. Though the exact recovery of an almost sparse signal
${\mbi {x}}$
is no longer feasible, the main contribution reveals that the support set of the best
$k$
-term approximation of
${\mbi {x}}$
can be recovered under reasonable conditions. The error bound between
${\mbi {x}}$
and the estimation of OMP is also derived. By constructing an example it is also demonstrated that the sufficient conditions for support recovery of the best
$k$
-term approximation of
${\mbi {x}}$
are rather tight. When
${\mbi {x}}$
is strong-decaying, it is proved that the sufficient conditions for support recovery of the best
$k$
-term approximation of
${\mbi {x}}$
can be relaxed, and the support can even be recovered in the order of the entries' magnitude. Our results are also compared in detail with some related previous ones.