On Sparse Solutions of Underdetermined Linear Systems

We first explain the research problem of finding the sparse solution of underdetermined linear systems with some applications. Then we explain three different approaches how to solve the sparse solution: the l1 approach, the orthogonal greedy approach, and the lq approach with 0 < q ≤ 1. We mainly survey recent results and present some new or simplified proofs. In particular, we give a good reason why the orthogonal greedy algorithm converges and why it can be used to find the sparse solution. About the restricted isometry property (RIP) of matrices, we provide an elementary proof to a known result that the probability that the random matrix with iid Gaussian variables possesses the PIP is strictly positive. 1 The Research Problem Given a matrix Φ of size m× n with m ≤ n, let Rk = {Φx,x ∈ R, ‖x‖0 ≤ k} be the range of Φ of all the k-component vectors, where ‖x‖0 stands for the number of the nonzero components of x.