Fast Iterative Hard Thresholding for Compressed Sensing

Algebraic Pursuit (ALPS) is an effective class of iterative hard thresholding algorithms for compressed sensing, with 1-ALPS(2) being the most computationally efficient variant of ALPS. We present a proof of convergence, using restricted isometry constants, for 1-ALPS(2) as well as the recently introduced Fast Iterative Hard Thresholding (FIHT). Large scale empirical testing shows FIHT is superior to 1-ALPS(2) in terms of both the sizes of the problems that are recoverable and overall computational time.