A Non-Iterative Procedure for Computing Sparse and Sparsifiable Solutions to Slightly Underdetermined Linear Systems of Equations

The problem of computing sparse (mostly zero) solutions to underdetermined linear systems of equations has received much attention recently, due to its applications to compressed sensing. Under mild assumptions, the sparsest solution has minimum-L1norm, and can be computed using linear programming. In some applications (valid deconvolution, singular linear transformations), the linear system is underdetermined by a relatively small amount, and a simpler solution is desirable. This paper presents a closed-form solution for computing the K-sparse solution to an M-by-N underdetermined linear system of equations, if N exceeds (K+1)(N-M+1). A numerical example and program illustrates the new algorithm. Keywords— Sparse reconstruction Phone: 734-763-9810. Fax: 734-763-1503. Email: [email protected]. EDICS: 2-REST.