  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.
