A Unifying Analysis of Projected Gradient Descent for ℓ_p-constrained Least Squares
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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.
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