Gradient descent with sparsification

Abstract

We present an algorithm for finding an s-sparse vector x that minimizes the square-error y - Φx2 where Φ satisfies the restricted isometry property (RIP), with isometric constant δ2s 1 and Hs sets all but s largest magnitude coordinates to zero. GraDeS converges to the correct solution in constant number of iterations. The condition δ2s < 1/3 is most general for which a near-linear time algorithm is known. In comparison, the best condition under which a polynomial-time algorithm is known, is δ2s < 2 - 1. Our Matlab implementation of GraDeS outperforms previously proposed algorithms like Subspace Pursuit, StOMP, OMP, and Lasso by an order of magnitude. Curiously, our experiments also uncovered cases where L1-regularized regression (Lasso) fails but GraDeS finds the correct solution.