The bounds of restricted isometry constants for low rank matrices recovery

This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 〈 p ≤ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant （RIC）. Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ4Ar 〈 0.558 and δ3rA 〈 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ2rA 〈 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ2tA 〈 0.4931 and δrA 〈 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ2rA 〉1√2 or δrA 〉 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ2rA and δrA.Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p （0 〈 p 〈 1） quasi norm minimization problem.

Stable Recovery of Low Rank Matrices From Nuclear Norm Minimization

Low rank matrix recovery is a new topic drawing the attention of many researchers which addresses the problem of recovering an unknown low rank matrix from few linear measurements. The matrix Dantzig selector and the matrix Lasso are two important algorithms based on nuclear norm minimization. In this paper, we first prove some decay properties of restricted isometry constants, then we discuss the recovery errors of these two algorithms and give a new bound of restricted isometry constant to guarantee stable recovery, which improves the results of [11].

New Bounds for RIC in Compressed Sensing

This paper gives new bounds for restricted isometry constant(RIC)in compressed sensing.LetΦbe an m×n real matrix and k be a positive integer with k≤n.The main results of this paper show that if the restricted isometry constant ofΦsat-isfiesδ8ak<1 andδk+ak<3/2−1+√(4a+3)^(2)−8/8aforα>3/8,then k-sparse solution can be recovered exactly via l1 minimization in the noiseless case.In particular,whenα=1,1.5,2 and3,we haveδ2k<0.5746 andδ8k<1,orδ2.5k<0.7046 andδ12k<1,orδ3k<0.7731 andδ16k<1 orδ4k<0.8445 andδ24k<1.

Stable recovery of low-rank matrix via nonconvex Schatten p-minimization

In this paper, a sufficient condition is obtained to ensure the stable recovery(ε≠ 0) or exact recovery(ε = 0) of all r-rank matrices X ∈ Rm×nfrom b = A(X) + z via nonconvex Schatten p-minimization for anyδ4r∈ [3～(1/2))2, 1). Moreover, we determine the range of parameter p with any given δ4r∈ [(3～(1/2))/22, 1). In fact, for any given δ4r∈ [3～(1/2))2, 1), p ∈(0, 2(1- δ4r)] suffices for the stable recovery or exact recovery of all r-rank matrices.

THE EXACT RECOVERY OF SPARSE SIGNALS VIA ORTHOGONAL MATCHING PURSUIT

This paper aims to investigate sufficient conditions for the recovery of sparse signals via the orthogonal matching pursuit （OMP） algorithm. In the noiseless case, we present a novel sufficient condition for the exact recovery of all k-sparse signals by the OMP algorithm, and demonstrate that this condition is sharp. In the noisy case, a sufficient condition for recovering the support of k-sparse signal is also presented. Generally, the computation for the restricted isometry constant （RIC） in these sufficient conditions is typically difficult, therefore we provide a new condition which is not only computable but also sufficient for the exact recovery of all k-sparse signals.