Narrow Results

In this paper, we give an application of the perturbation inequality to the low rank matrix recovery problem and provide a condition on the linear map of underdetermined linear system that every minimal rank symmetric matrix $X\in {ℝ}^{m\times n}$ can be exactly recovered from the linear measurement $b={𝒜}(X)\in {ℝ}^q$ via some Schatten $p_0$ norm minimization. Moreover it is shown that the explicit bound on exponent $p_0$ in the Schatten $p_0$ norm minimization can be exactly extracted.

Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in compressed sensing, the two models are equivalent with the central distinctness that the noise associated with double noise model is larger by a factor to $mn/M$, where $m,n$ are the dimensions of the matrix and $M$ is the number of measurements. Additionally, this paper discusses the reconstruction of low-rank matrices in the setting, presents sufficient conditions based on the associating null space property to guarantee the robust recovery and obtains the number of measurements. Furthermore, for the non-Gaussian noise scenario, we further explore it and give the corresponding result. The simulation experiments conducted, on the one hand show effect of noise variance on recovery performance, on the other hand demonstrate the verifiability of the proposed model.

In practical examples, there are numerous signals that are sparse in a redundant frame rather than an orthonormal basis. This paper mainly focuses on such sparse recovery via $l_q$-analysis-based dual frame with Weibull matrices under the assumption that signals are sparse or compressible in a general frame. First, we give the $l_{2,q}$ robust null space property and show that it is weaker than the $D$-RIP when $D$ is a general frame. Second, we show that Weibull random matrices with a general dual frame satisfy the $l_{2,q}$ robust null space property with high probability. Finally, we investigate the stability estimate of $l_q$-analysis based dual frame with Weibull matrices. The result shows that it remains stable and can guarantee accurate recovery of signals with high probability.

This paper first discusses the relationship between the rank null space property (NSP) and the nuclear norm minimization. Several versions of the rank NSP, i.e. the stable rank NSP, robust rank NSP and Frobenius robust rank NSP are proposed, and their equivalent forms are derived. At the same time, it is shown that the stable rank NSP is weaker than the rank restricted isometry property (RIP) to recover the low-rank matrices via the nuclear norm minimization. Finally, the rank NSP is extended to the case of Schatten-$q$ NSP for $0<q<1$, and the solutions to the Schatten-$q$ quasi-norm minimization are characterized by the different types of Schatten-$q$ NSP.

This paper studies a kind of weighted $l_1$-minimization that is motivated by function interpolation. Combining with the weighted robust null space property, we first propose a new sufficient condition for robust recovery via the weighted $l_1$-minimization when the measurements are corrupted by arbitrary noise without requiring the proper estimation of noise level. Second, we investigate the instance optimality of the weighted $l_1$-minimization decoder according to the weighted quotient property and the weighted restricted isometry property (RIP). In addition, to give a better error estimation in a general problem to recover noisy compressible signals, we improve the $\omega$-RIP constant ${\delta }_{\omega ,3s}$ from $1/3$ to $(\sqrt{17}-1)/8$ by using the weighted sparsity and Stechkin-type estimate. Our results show that the weighted $l_1$-minimization remains not only stable but also robust to reconstruct signals with noisy observations.

The null space property (NSP), which relies merely on the null space of the sensing matrix column space, has drawn numerous interests in sparse signal recovery. This paper studies NSP of the weighted ${\ell }_r-{\ell }_1$ ($r\in (0,1]$) minimization. Several versions of NSP of the weighted ${\ell }_r-{\ell }_1$ minimization including the weighted ${\ell }_r-{\ell }_1$ NSP, the weighted ${\ell }_r-{\ell }_1$ stable NSP, the weighted ${\ell }_r-{\ell }_1$ robust NSP and the ${\ell }_q$ weighted ${\ell }_r-{\ell }_1$ robust NSP for $1\le q\le 2$, are proposed, as well as the associating considerable results are derived. Under these NSPs, sufficient conditions for the recovery of (sparse) signals with the weighted ${\ell }_r-{\ell }_1$ minimization are established. Furthermore, we show that to some extent, the weighted ${\ell }_r-{\ell }_1$ stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou (2022).

This paper introduces a nonconvex ${\ell }_{1-2}$-analysis model: