- RSP-Based Analysis for Sparsest and Least $\ell_1$-Norm Solutions to Underdetermined Linear Systems Y Zhao(y.zhao.2bham.ac.uk) Abstract: Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does $\ell_1$-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary and sufficient condition for the uniqueness of least $\ell_1$-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least $\ell_1$-norm solution to a linear system if and only if the so-called \emph{range space property} (RSP) holds at this solution. This yields a broad understanding of the relationship between $\ell_0$- and $\ell_1$-minimization problems. Our analysis indicates that the RSP truly lies at the heart of the relationship between these two problems. Through RSP-based analysis, several important questions in this field can be largely addressed. For instance, how to efficiently interpret the gap between the current theory and the actual numerical performance of $\ell_1$-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does $\ell_1$-minimization guarantee to find one of them? Moreover, new matrix properties (such as the \emph{RSP of order $K$} and the \emph{Weak-RSP of order $K$}) are introduced in this paper, and a new theory for sparse signal recovery based on the RSP of order $K$ is established. Keywords: Underdetermined linear system, sparsest solution, least $\ell_1$-norm solution, range space property, strict complementarity, sparse signal recovery, compressed sensing. Category 1: Linear, Cone and Semidefinite Programming Category 2: Linear, Cone and Semidefinite Programming (Linear Programming ) Category 3: Global Optimization Citation: Download: [PDF]Entry Submitted: 07/17/2013Entry Accepted: 07/17/2013Entry Last Modified: 07/17/2013Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.