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 ℓ p -relaxation, orthogonal matching pursuit (OMP), and orthogonal least squares (OLS). Our result is based on the coherence μ of the dictionary and relaxes the well-known condition μ