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In this paper, we investigate the low-rank matrix recovery problem from linear observations. Inspired by the matrix Lasso and Dantzig selector, we propose nuclear norm regularized models with two data fitting items: ℓ 2 -loss function and Dantzig selector. We establish the recovery errors for both constrained and unconstrained models under the matrix restricted isometry property frame. For the unconstrained model, we obtain a tighter error bound compared with the previous study by W. Wang et al. (2021) under certain conditions. Furthermore, we design an effective solving algorithm for the proposed unconstrained model. Our numerical experiments demonstrate that our method outperforms the Lasso and Dantzig selector models. • Stable recovery errors for nuclear norm regularized models are established. • Efficient algorithm based on PSS method for low-rank matrix recovery is proposed. • Our method achieves good recovery performance in numerical experiments. [ABSTRACT FROM AUTHOR]

In numerous applications, such as DNA microarrays, face recognition, and spectral unmixing, we need to acquire a non-negative K -sparse signal x from an underdetermined linear model y = A x + v , where A is a sensing matrix and v is a noise vector. In this paper, we first propose a ReLU-based hard-thresholding algorithm (RHT) to recover x by taking advantage of its non-negative sparsity. Two sufficient conditions for stable recovery with RHT are then developed, which are respectively based on the restricted isometry property (RIP) and mutual coherence of the sensing matrix A. As far as we know, these two sufficient conditions are the best for hard-thresholding-type algorithms. Numerical experiments show that RHT has better overall recovery performance in the recovery non-negative sparse signals than the non-negative least squares (NNLS) algorithm, some hard-thresholding-type algorithms including the iterative hard-thresholding (IHT) algorithm, hard-thresholding pursuit (HTP), Newton-step-based iterative hard-thresholding algorithm (NSIHT), and Newton-step-based hard-thresholding pursuit (NSHTP), several non-negative sparse recovery approaches including non-negative orthogonal matching pursuit (NNOMP), fast NNOMP (FNNOMP), non-negative orthogonal least squares (NNOLS), and non-negative regularization (NNREG) method. In terms of efficiency, RHT has a similar performance to IHT, and is much more efficient than other tested algorithms. • Efficient hard-thresholding-type algorithm for non-negative sparse recovery. • Sufficient conditions for stable recovery with RHT based on RIP and mutual coherence. • RHT achieves good recovery performance in both stimulated and real-world experiments. • RHT shows similar efficiency with IHT, and outperforms the remaining algorithms. [ABSTRACT FROM AUTHOR]