Iteratively reweighted least squares minimization for sparse recovery

• Published in: Communications on pure and applied mathematics Vol. 63; no. 1; pp. 1 - 38
• Main Authors: Daubechies, Ingrid, DeVore, Ronald, Fornasier, Massimo, Güntürk, C. Si̇nan
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.01.2010; Wiley; John Wiley and Sons, Limited
• Subjects: Calculus of variations and optimal control; Convergence; Exact sciences and technology; Linear programming; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Optimization algorithms; Sciences and techniques of general use; Vector space; Least squares method; Optimization method; Linear programming; Iteration; Fast algorithm; Hyperplane; Method study
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20303

Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ℝN that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx 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 𝓁1‐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 𝓁2(w)‐norm. If x(n) is the solution at iteration step n, then the new weight w(n) is defined by w i(n) := [|x i(n)|2 + ε n2]−1/2, i = 1, …, N, for a decreasing sequence of adaptively defined εn; this updated weight is then used to obtain x(n + 1) and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x(n) converges for all y, regardless of whether Φ−1(y) contains a sparse vector. If there is a sparse vector in Φ−1(y), then the limit is this sparse vector, and when x(n) is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w i(n) = [|x i(n)|2 + ε n2]−1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc.