A new sufficient condition for sparse vector recovery via ℓ1 − ℓ2 local minimization

In this paper, we provide a necessary condition and a sufficient condition such that any [Formula: see text]-sparse vector [Formula: see text] can be recovered from [Formula: see text] via [Formula: see text] local minimization. Moreover, we further verify that the sufficient condition is naturally valid when the restricted isometry constant of the measurement matrix [Formula: see text] satisfies [Formula: see text]. Compared with the existing [Formula: see text] local recoverability condition [Formula: see text], this result shows that [Formula: see text] local recoverability contains more measurement matrices.

Guarantees of Fast Band Restricted Thresholding Algorithm for Low-Rank Matrix Recovery Problem

Affine matrix rank minimization problem is a famous problem with a wide range of application backgrounds. This problem is a combinatorial problem and deemed to be NP-hard. In this paper, we propose a family of fast band restricted thresholding (FBRT) algorithms for low rank matrix recovery from a small number of linear measurements. Characterized via restricted isometry constant, we elaborate the theoretical guarantees in both noise-free and noisy cases. Two thresholding operators are discussed and numerical demonstrations show that FBRT algorithms have better performances than some state-of-the-art methods. Particularly, the running time of FBRT algorithms is much faster than the commonly singular value thresholding algorithms.

Perturbation Analysis of Orthogonal Least Squares

The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

Sparse signal recovery via non-convex optimization and overcomplete dictionaries

In this paper, we address the problem of recovering signals from undersampled data where such signals are not sparse in an orthonormal basis, but in an overcomplete dictionary. We show that if the combined matrix obeys a certain restricted isometry property and if the signal is sufficiently sparse, the reconstruction that relies on [Formula: see text] minimization with [Formula: see text] is exact. In addition, under a mild assumption about the dictionary [Formula: see text], we use a similar method [H. Rauhut et al., Compressed sensing and redundant dictionaries, IEEE Trans. Inf. Theory 54(5) (2008) 2210–2219] to derive an estimation of the restricted isometry constant of the composed matrix [Formula: see text]. Finally, the performance of the [Formula: see text] minimization is testified by some numerical examples.

A Sharp Bound on RIC in Generalized Orthogonal Matching Pursuit

The generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of the orthogonal matching pursuit (OMP). It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every K-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) δNK+1 of the sensing matrix A satisfiesthen the gOMP can perfectly recover every K-sparse signal x from y = Ax. Furthermore, the bound is proved to be sharp. In the noisy case, the above bound on RIC combining with an extra condition on the minimum magnitude of the nonzero components of K-sparse signals can guarantee that the gOMP selects all of the support indices of the K-sparse signals.