Sparse Recovery Conditions and Performance Bounds for $\ell _p$-Minimization

Sparse Recovery Analysis of Generalized $J$-Minimization with Results for Sparsity Promoting Functions with Monotonic Elasticity

“…P f (3) with f (x) = |x| p ), for p ∈ (0, 1]. Recently, the work [23] has addressed the issue of both determining an upper bound on the J−NSC for the problem P p , p ∈ (0, 1], with explicit dependence on δ 2K , K, p, as well as finding upper bounds on the RIC δ 2K that ensures exact recovery of a K-sparse signal by solving the problem P p . Furthermore, [23] has produced the sharpest of all the RIC bounds derived so far.…”

“…Recently, the work [23] has addressed the issue of both determining an upper bound on the J−NSC for the problem P p , p ∈ (0, 1], with explicit dependence on δ 2K , K, p, as well as finding upper bounds on the RIC δ 2K that ensures exact recovery of a K-sparse signal by solving the problem P p . Furthermore, [23] has produced the sharpest of all the RIC bounds derived so far. However, in recent years, many researchers have proposed and worked with many other non-convex sparsity promoting functions, for example, the concave exponential f (x) = 1 − e −|x| p , p ∈ [0, 1] [24, 25,26], the Lorentzian function f (x) = ln(1 + |x| p ) [27,28],…”

“…1. Can the arguments of [23] be extended to the case of general sparsity promoting functions to derive bounds on NSC for the problem P f (3)?…”

“…In this paper, we answer the above questions related to the performance of sparse recovery generalized nonconvex minimization. First, after describing the notations adopted in the paper as well as some Lemmas and definitions that are used throughout the rest of the paper in Section 2, we present the main technical tool in Section 3, where we extend the techniques of [23] to general sparsity promoting functions. Second, we use this tool to present an upper bound on the null space constant associated with the generalized problem P f in Section 3.…”

Stability of 1-Bit Compressed Sensing in Sparse Data Reconstruction

“…Moreover, some added assumptions have to be added on the measurement matrix A to ensure that a sparse solution/signal could be exactly recovered by l 1 minimization. ese conditions include restricted isometry property [9][10][11], coherence condition [12], null space property [8,13,14], and range space property [15,16]. In recent research, some work has been done concerning the robust reconstruction condition (RRC) based on the above traditional properties and their variants, e.g., exact reconstruction condition [17], double null space property [18], and null space property [19].…”