Faster & greedier: algorithms for sparse reconstruction of large datasets

We consider the problem of performing sparse reconstruction of large-scale data sets, such as the image sequences acquired in dynamic MRI. Here, both conventional L₁ minimization through interior point methods and orthogonal matching pursuit (OMP) are not practical. Instead we present an algorithm that combines fast directional updates based around conjugate gradients with an iterative thresholding step similar to that in StOMP but based upon a weak greedy selection criterion. The algorithm can achieve OMP-like performance and the rapid convergence of StOMP but with MP-like complexity per iteration. We also discuss recovery conditions applicable to this algorithm.