Stepwise Optimal Subspace Pursuit for Improving Sparse Recovery

We propose a new iterative algorithm to reconstruct an unknown sparse signal x from a set of projected measurements y = Φx . Unlike existing methods, which rely crucially on the near orthogonality of the sampling matrix Φ , our approach makes stepwise optimal updates even when the columns of Φ are not orthogonal. We invoke a block-wise matrix inversion formula to obtain a closed-form expression for the increase (reduction) in the L
₂
-norm of the residue obtained by removing (adding) a single element from (to) the presumed support of x . We then use this expression to design a computationally tractable algorithm to search for the nonzero components of x . We show that compared to currently popular sparsity seeking matching pursuit algorithms, each step of the proposed algorithm is locally optimal with respect to the actual objective function. We demonstrate experimentally that the algorithm significantly outperforms conventional techniques in recovering sparse signals whose nonzero values have exponentially decaying magnitudes or are distributed N(0,1) .