Gradient Descent with Sparsification: An Iterative Algorithm for Sparse Recovery with Restricted Isometry Property

In this paper, we present an algorithm for finding an s-sparse vector x that minimizes the square-error ∥y − Φx∥ 2 where Φ satisfies the restricted isometry property (RIP). Our algorithm, called GraDeS (Gradient Descent with Sparsification) starts from an arbitrary s-sparse x and iteratively updates it as: x← Hsx + 1γ · Φt (y − Φx)where γ > 1 is a constant and Hs sets all but largest s coordinates in absolute value to zero. We show that GraDeS, in constant number of iterations, computes the correct s-sparse solution to the system y = Φx where Φ satisfies the condition that the isometric constant δ2s < 1/3. This is the most general condition for which, near-linear time algorithm is known. In comparison, the best condition under which any polynomial-time algorithm is known, is δ2s < √2 − 1. An important contribution of the paper is to analyze how the hard-thresholding function Hs acts w.r.t. the potential ∥y − Φx∥ 2 . A special case of GraDeS, corresponding to γ = 1, called Iterative Hard Thresholding (IHT), was previously shown to converge when δ3s < 1/√32. Our Matlab implementation of GraDeS out-performs previously proposed algorithms like Subspace Pursuit, StOMP, OMP, and Lasso by an order of magnitude. Curiously, our experiments also uncovered several cases where L1-regularized regression (Lasso) fails but GraDeS finds the correct solution.