TUNING-FREE JOINT SPARSE RECOVERY VIA OPTIMIZATION TRANSFER Evgenia Chunikhina, Gregory Gutshall, Raviv Raich, and Thinh Nguyen Department of EECS,Oregon State University, Corvallis, Oregon 97331 {chunikhe,gutshalg,raich,thinhq}eecs.oregonstate.edu ABSTRACT Multiple measurement vector (MMV) problem addresses the recovery of a set of sparse vectors that have common sparsity pattern. In this paper, we consider a variant of the MMV problem where the common sparsity pattern is obfuscated by an additive noise. Specifically, we study the conditions for perfect reconstruction of the original sparsity pattern. Based on these, we develop a tuning-free algorithm for re- covering jointly sparse solutions via the transfer optimization approach. We provide a preliminary numerical evaluation to illustrate our approach. Index Terms— Sparse representation, joint sparsity, multiple-measurement vector (MMV), optimization trans- fer 1. INTRODUCTION The problem of computing sparse solutions for linear inverse problems has received notable attention in the last years, es- pecially in the signal processing community. Many different methods have been proposed to solve this problem [1] and [2]. In order to enhance the recoverability, additional infor- mation about the underlying solution structure, such as group sparsity, can be taken into account. In our work, we consider a special case of the group spar- sity structure, which is a problem of reconstruction of jointly sparse solutions, also known as the multiple measurement vector (MMV) problem. Jointly sparse solutions share the same nonzero support and appear in many applications, such as distributed compressive sensing, source localization, and magnetic resonance imaging. Our interest was motivated by the problem of deconvolving the threshold ionization ener- gies of a measurement recorded from a Resonant Electron Capture-Time of Flight (REC-ToF) mass spectrometer. Theo- retically, a fragment ion has distinct sparse ionization thresh- old energies. However, the measured ionization curve is con- voluted by the electron energy distribution of the ionization source, which may be characterized through measurement. Most notable among the methods that have been devel- oped to solve the MMV problem are the forward sequential This work was partially supported by the Air Force Office of Scientific Research grant number FA9550-09-1-0471 for R.R. and E.C. and by the De- partment of Analytical Chemistry and the Mass Spectrometry faculty grant NIEHS (P30 ES00210) for G.G. search-based method [3], ReMBo (which, reduces a MMV problem to a set of singular measurement vector problems) [4], a method based upon the alternating direction principle [5], along with many others. The variation of problems that allow the presence of noise was recently studied in [2], [6]. Our work is focused on finding the jointly sparse solution to the MMV problem whose the original sparsity pattern has been obfuscated by an additive noise. We study the condi- tions for perfect reconstruction of the original sparsity pat- tern. Based on these, we propose a tuning-free algorithm for recovering jointly sparse solutions via the optimization trans- fer approach. We provide a preliminary numerical evaluation of our approach. 2. PROBLEM FORMULATION In this paper we define matrices by uppercase letters and vec- tors by lowercase letters. For a matrix X, x i indicates its i-th row and x j indicates its j -th column. Let ‖X‖ R 0 denote the number of rows of matrix X that have non-zero elements, i.e., ‖X‖ R 0 = card{i|‖X T e i ‖ 2 = 0}, where e i the canonical vector satisfying e i (j )=1 for j = i and 0 otherwise. The multiple measurement vector (MMV) problem can be formulated as minimize X ‖X‖ R 0 subject to n  l=1 ‖A l x l - y l ‖ 2 2  ǫ, (1) where the mixing matrices A l ∈ R m×k may be different ∀l = 1,n, and the solution vectors x l ∈ R k×1 and measurement vectors y l ∈ R m×1 are such that y l = A l x 0l + ξ l , with vector ξ l =[ξ l1 ,ξ l2 ,...,ξ ln ] T representing the additive noise ∀l = 1,n and X 0 ∈ R k×n is the original s 0 row-sparse matrix we are interested in recovering. Here, n  1 is the number of measurement vectors. We assume n ≪ m. Matrices A l ∈ R m×k are known and ob- tained from the physics of the problem. Without loss of gen- erality we assume that rank(A l )= m, m ≪ k, ∀l = 1,n. The main aim is to obtain the solution matrix X such that each solution vector x l is sparse and all solution vectors have a common sparsity pattern, i.e., the indices of the nonzero elements of x l must be the same ∀l = 1,n. 1913 978-1-4673-0046-9/12/$26.00 ©2012 IEEE ICASSP 2012