Iteratively reweighted least squares minimization for sparse recovery

Abstract Under certain conditions (known as the restricted isometry property , or RIP) on the m × N matrix Φ (where m < N ), vectors x ∈ ℝ N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φ x even though Φ −1 ( y ) is typically an ( N − m )—dimensional hyperplane; in addition, x is then equal to the element in Φ −1 ( y ) of minimal 𝓁 1 ‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x , as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w , the element in Φ −1 ( y ) with smallest 𝓁 2 ( w )‐norm.…