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 $\ell_{p}$-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 $0\leq p\leq1$, 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.