Matching Pursuit LASSO Part I: Sparse Recovery Over Big Dictionary

Large-scale sparse recovery (SR) by solving ℓ₁-norm relaxations over Big Dictionary is a very challenging task. Plenty of greedy methods have therefore been proposed to address big SR problems, but most of them require restricted conditions for the convergence. Moreover, it is non-trivial for them to incorporate the ℓ₁-norm regularization that is required for robust signal recovery. We address these issues in this paper by proposing a Matching Pursuit LASSO (MPL) algorithm, based on a novel quadratically constrained linear program (QCLP) formulation, which has several advantages over existing methods. Firstly, it is guaranteed to converge to a global solution. Secondly, it greatly reduces the computation cost of the ℓ₁-norm methods over Big Dictionaries. Lastly, the exact sparse recovery condition of MPL is also investigated.