The weighted$\ell _{r}-\ell _{1}$minimization with weight$\alpha$has been extensively employed to robustly estimate a high-dimensional sparse signal$x$coded by the underdetermined linear measurements$y=Ax+z$, where$A$and$z$are the measurement matrix and noise, respectively. In this paper, we demonstrate that if the restricted isometry constant (RIC)$\delta _{s}$of$A$fulfills\begin{align*} \delta _{s}< 1/\left(1+3t/\sqrt{5}\right), \end{align*}where$t$relies on sparsity level$s$for known model parameters$\alpha$and$r$, then any sparse signal$x$are ensured to be robustly reconstructed through solving the weighted$\ell _{r}-\ell _{1}$minimization in the noisy situation. The gained condition is testified to be much better that the state-of-art ones.

In this article, we bring forward a completely perturbed nonconvex Schatten p -minimization to address a model of completely perturbed low-rank matrix recovery (LRMR). This article based on the restricted isometry property (RIP) and the Schatten- p null space property (NSP) generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, and it gives the RIP condition and the Schatten- p NSP assumption that guarantee the recovery of low-rank matrix and the corresponding reconstruction error bounds. In particular, the analysis of the result reveals that in the case that p decreases 0 and for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition (Recht et al., 2010). In addition, we study the connection between RIP and Schatten- p NSP and discern that Schatten- p NSP can be inferred from the RIP. The numerical experiments are conducted to show better performance and provide outperformance of the nonconvex Schatten p -minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.

In this work, a novel self-powered visible light communication and positioning (SPVLCP) system is constructed, which can realize communication, positioning, and energy harvesting simultaneously. The resource allocation scheme applying orthogonal frequency division multiple (OFDM) modulation and the power splitting (PS) method is proposed for the SPVLCP system. Based on this scheme, a corresponding optimization allocation strategy is provided, aiming to maximize the data rate by optimizing the PS factor and the power allocated to subcarriers while ensuring positioning accuracy, harvested energy, and the LED transmit power. To solve the optimization issue, the successive convex approximation (SCA) algorithm and a series of equivalent substitutions are applied to convert the optimization issue from non-convex to convex. The findings demonstrated that the optimization allocation strategy can solve the trade-off problem among communication, positioning, and energy harvesting better than the uniform allocation strategy. Furthermore, the SCA-based algorithm has an obvious superiority over the conventional heuristic algorithm in solving the optimization issue for the proposed resource allocation scheme of the integrated SPVLCP system.

A plethora of previous studies indicates that making full use of multifarious intrinsic properties of primordial data is a valid pathway to recover original images from their degraded observations. Typically, both low-rankness and local-smoothness broadly exist in real-world tensor data such as hyperspectral images and videos. Modeling based on both properties has received a great deal of attention, whereas most studies concentrate on experimental performance, and theoretical investigations are still lacking. In this paper, we study the tensor compressive sensing problem based on the tensor correlated total variation, which is a new regularizer used to simultaneously capture both properties existing in the same dataset. The new regularizer has the outstanding advantage of not using a trade-off parameter to balance the two properties. The obtained theories provide a robust recovery guarantee, where the error bound shows that our model certainly benefits from both properties in ground-truth data adaptively. Moreover, based on the ADMM update procedure, we design an algorithm with a global convergence guarantee to solve this model. At last, we carry out experiments to apply our model to hyperspectral image and video restoration problems. The experimental results show that our method is prominently better than many other competing ones. Our code and Supplementary Material are available at https://github.com/fsliuxl/cs-tctv.

Recovering low-rank matrices by nuclear norm minimization and local-smooth matrices by total variation seminorm minimization are two common methods in the context of compressive sensing. As a matter of fact, the two properties simultaneously exist in many real-world datasets, typically exampling hyperspectral images. The two methods may not perform well in this situation. To better address this issue, in this letter, we study the correlated total variation norm minimization problem both theoretically and numerically. We obtain an error bound for the robust recovery of our method in theory, which reflects that this model indeed benefits from low-rank and local-smooth properties of the matrix to be restored. Experiments on the recovery of hyperspectral images show that this model is superior to many other competing ones.

Mining structural priors in data is a widely recognized technique for hyperspectral image (HSI) denoising tasks, whose typical ways include model-based methods and data-based methods. The model-based methods have good generalization ability, while the runtime can hardly meet the fast processing requirements of the practical situations due to the large size of an HSI${\mathbf {X}}\in \mathbb {R}^{\textrm {MN}\times B}$. For the data-based methods, they perform relatively fast on new test data once they have been trained. However, their generalization ability is always insufficient. In this article, we propose a fast model-based approach via a novel regularizer named the representative coefficient total variation (RCTV) to simultaneously characterize the low-rank and local smooth properties. The RCTV regularizer is proposed based on the observation that the representative coefficient matrix${\mathbf {U}}\in \mathbb {R}^{\textrm {MN}\times R} (R\ll B)$obtained by orthogonally transforming the original HSI${\mathbf {X}}$can inherit the strong local-smooth prior of${\mathbf {X}}$. Since$R/B$is very small, the model based on the RCTV regularizer has lower time complexity. In addition, we find that the representative coefficient matrix${\mathbf {U}}$is robust to noise, and thus, the RCTV regularizer can somewhat promote the robustness of the HSI denoising model. Extensive experiments on mixed noise removal demonstrate that the proposed method realizes a perfect compromise between denoising performance and denoising speed compared with other state-of-the-art methods. Remarkably, the denoising speed of our proposed method outperforms all competing model-based techniques and is comparable with the deep learning-based approaches. The code of our algorithm is released athttps://github.com/andrew-pengjj/rctv.git.

The tensor-tensor product-induced tensor nuclear norm (t-TNN) (Lu et al., 2020) minimization for low-tubal-rank tensor recovery attracts broad attention recently. However, minimizing the t-TNN faces some drawbacks. For example, the obtained solution could be suboptimal to the original problem due to its loose approximation. In this article, we extract a unified nonconvex surrogate of the tensor tubal rank as a tighter regularizer, which involves many popular nonconvex penalty functions. An iterative reweighted t-TNN algorithm is proposed to solve the resulting generalized nonconvex tubal rank minimization for tensor recovery. It converges to a critical point globally with rigorous proofs based on the Kurdyka-Łojasiwicz property. Furthermore, we provide the theoretical guarantees for exact and robust recovery by developing the tensor null space property. Extensive experiments demonstrate that our approach markedly enhances recovery performance compared with several state-of-the-art convex and nonconvex methods.