RSP-Based Analysis for Sparsest and Least $\ell_1$-Norm Solutions to Underdetermined Linear Systems

Abstract

Abstract-Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does ℓ1-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 ℓ1-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least ℓ1-norm solution to a linear system if and only if the so-called range space property (RSP) holds at this solution. This yields a broad understanding of the relationship between ℓ0-and ℓ1-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 ℓ1-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does ℓ1-minimization guarantee to find one of them? Moreover, new matrix properties (such as the RSP of order K and the 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.Index Terms-Underdetermined linear system, sparsest solution, least ℓ1-norm solution, range space property, strict complementarity, sparse signal recovery, compressed sensing.

Block sparse vector recovery via weighted generalized range space property

“…The problems (PL0) and (PL1) are said to be strictly equivalent if they both have a unique solution and the two solutions coincide [21]. When the problem (PL0) has multiple solutions, and the solution of (PL1) coincides with one of the solutions of (PL0), then we say that (PL0) and (PL1) are equivalent [21].…”

“…When the problem (PL0) has multiple solutions, and the solution of (PL1) coincides with one of the solutions of (PL0), then we say that (PL0) and (PL1) are equivalent [21].…”

“…the Range Space Property (RSP) of order K [21]. In practice however, those conditions may not be met, which means that the least 1 -norm solution and the least 0 -norm solution are not the same.…”

Generalized range space property for group sparsity of linear underdetermined systems

“…The problems (PL0) and (PL1) are said to be strongly equivalent if each has a unique solution and the two solutions coincide [1]. The conditions which the sensing matrix A should satisfy for strong equivalence between (PL0) and (PL1) include the Restricted Isometry Property (RIP) [2], the Null Space Property (NSP) [3] and the Mutual Coherence [4].…”

A Smoothing Method for Sparse Optimization over Polyhedral Sets

“…Moreover, several criteria have been found which guarantee that solving (P 1 ) will also solve (P 0 ) under various assumptions involving the coecients of the matrix A. These criteria, denoted mutual coherence [25], restricted isometry property [14], null space property [19], exact recovery condition [59,31], and the range space property [66], show the eciency of this convex approximation to solve (P 0 ).…”