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.
CITATION STYLE
Kozlov, I., & Petukhov, A. (2010). Sparse Solutions of Underdetermined Linear Systems. In Handbook of Geomathematics (pp. 1243–1259). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-01546-5_42
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