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

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

2011-07-22

arxiv.org/abs/1107.4623v4

Mathematics

Numerical Analysis

16 pages, 3 Figures. The projection algorithm and the corresponding theorem are withdrawn. The necessary conditions for a proj

Scientific paper

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.

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