Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Sparse signal reconstruction from limited data using FOCUSS: are-weighted minimum norm algorithm
IEEE Transactions on Signal Processing
Sparse representations in unions of bases
IEEE Transactions on Information Theory
On sparse representations in arbitrary redundant bases
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Just relax: convex programming methods for identifying sparse signals in noise
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Majorization–Minimization Algorithms for Wavelet-Based Image Restoration
IEEE Transactions on Image Processing
Stable recovery of sparse signals and an oracle inequality
IEEE Transactions on Information Theory
New bounds for restricted isometry constants
IEEE Transactions on Information Theory
Information Sciences: an International Journal
Unsupervised images segmentation via incremental dictionary learning based sparse representation
Information Sciences: an International Journal
Hi-index | 754.96 |
This paper investigates conditions under which the solution of an underdetermined linear system with minimal lP norm, 0 p ≤ 1, is guaranteed to be also the sparsest one. Matrices are constructed with restricted isometry constants (RIC) δ2m arbitrarily close to 1/ √2 ≅ 0.707 where sparse recovery with p = 1 fails for at least one m-sparse vector, as well as matrices with δ2m arbitrarily close to one where l1 minimization succeeds for any m-sparse vector. This highlights the pessimism of sparse recovery prediction based on the RIC, and indicates that there is limited room for improving over the best known positive results of Foucart and Lai, which guarantee that l1 minimization recovers all m-sparse vedors for any matrix with δ2m lp recovery (0 ≤ p ≤ 1) with matrices of unit spectral norm, which are expressed in terms of the minimal singular values of 2m-column submatrices. Compared to l1 minimization, lp minimization recovery failure is shown to be only slightly delayed in terms of the RIC values. Furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used. It is shown that when lp optimization is attempted using an iterative reweighted l1 scheme, failure can still occur for δ2m arbitrarily close to 1/ √2.