A Unifying Analysis of Projected Gradient Descent for $\ell_p$-constrained Least Squares

Abstract In this paper we study the performance of the Projected Gradient Descent (PGD) algorithm for l p -constrained least squares problems that arise in the framework of compressed sensing. Relying on the restricted isometry property, we provide convergence guarantees for this algorithm for the entire range of 0 ⩽ p ⩽ 1 , that include and generalize the existing results for the iterative hard thresholding algorithm and provide a new accuracy guarantee for the iterative soft thresholding algorithm as special cases. Our results suggest that in this group of algorithms, as p increases from zero to one, conditions required to guarantee accuracy become stricter and robustness to noise deteriorates.