This paper investigates the problem of recovering missing samples using methods based on sparse representation adapted especially for image signals. Instead of $l_2$-norm or Mean Square Error (MSE), a new perceptual quality measure is used as the similarity criterion between the original and the reconstructed images. The proposed criterion called Convex SIMilarity (CSIM) index is a modified version of the Structural SIMilarity (SSIM) index, which despite its predecessor, is convex and uni-modal. We derive mathematical properties for the proposed index and show how to optimally choose the parameters of the proposed criterion, investigating the Restricted Isometry (RIP) and error-sensitivity properties. We also propose an iterative sparse recovery method based on a constrained $l_1$-norm minimization problem, incorporating CSIM as the fidelity criterion. The resulting convex optimization problem is solved via an algorithm based on Alternating Direction Method of Multipliers (ADMM). Taking advantage of the convexity of the CSIM index, we also prove the convergence of the algorithm to the globally optimal solution of the proposed optimization problem, starting from any arbitrary point. Simulation results confirm the performance of the new similarity index as well as the proposed algorithm for missing sample recovery of image patch signals.

Inference and Estimation in Missing Information (MI) scenarios are important topics in Statistical Learning Theory and Machine Learning (ML). In ML literature, attempts have been made to enhance prediction through precise feature selection methods. In sparse linear models, LASSO is well-known in extracting the desired support of the signal and resisting against noisy systems. When sparse models are also suffering from MI, the sparse recovery and inference of the missing models are taken into account simultaneously. In this paper, we will introduce an approach which enjoys sparse regression and covariance matrix estimation to improve matrix completion accuracy, and as a result enhancing feature selection preciseness which leads to reduction in prediction Mean Squared Error (MSE). We will compare the effect of employing covariance matrix in enhancing estimation accuracy to the case it is not used in feature selection. Simulations show the improvement in the performance as compared to the case where the covariance matrix estimation is not used.

In this paper, we study the missing sample recovery problem using methods based on sparse approximation. In this regard, we investigate the algorithms used for solving the inverse problem associated with the restoration of missed samples of image signal. This problem is also known as inpainting in the context of image processing and for this purpose, we suggest an iterative sparse recovery algorithm based on constrained $l_1$-norm minimization with a new fidelity metric. The proposed metric called Convex SIMilarity (CSIM) index, is a simplified version of the Structural SIMilarity (SSIM) index, which is convex and error-sensitive. The optimization problem incorporating this criterion, is then solved via Alternating Direction Method of Multipliers (ADMM). Simulation results show the efficiency of the proposed method for missing sample recovery of 1D patch vectors and inpainting of 2D image signals.

In this letter, we propose an algorithm for recovery of sparse and low rank components of matrices using an iterative method with adaptive thresholding. In each iteration, the low rank and sparse components are obtained using a thresholding operator. This algorithm is fast and can be implemented easily. We compare it with one of the most common fast methods in which the rank and sparsity are approximated by $\ell_1$ norm. We also apply it to some real applications where the noise is not so sparse. The simulation results show that it has a suitable performance with low run-time.

In this paper, prediction for linear systems with missing information is investigated. New methods are introduced to improve the Mean Squared Error (MSE) on the test set in comparison to state-of-the-art methods, through appropriate tuning of Bias-Variance trade-off. First, the use of proposed Soft Weighted Prediction (SWP) algorithm and its efficacy are depicted and compared to previous works for non-missing scenarios. The algorithm is then modified and optimized for missing scenarios. It is shown that controlled over-fitting by suggested algorithms will improve prediction accuracy in various cases. Simulation results approve our heuristics in enhancing the prediction accuracy.

In this paper, we investigate the recovery of a sparse weight vector (parameters vector) from a set of noisy linear combinations. However, only partial information about the matrix representing the linear combinations is available. Assuming a low-rank structure for the matrix, one natural solution would be to first apply a matrix completion on the data, and then to solve the resulting compressed sensing problem. In big data applications such as massive MIMO and medical data, the matrix completion step imposes a huge computational burden. Here, we propose to reduce the computational cost of the completion task by ignoring the columns corresponding to zero elements in the sparse vector. To this end, we employ a technique to initially approximate the support of the sparse vector. We further propose to unify the partial matrix completion and sparse vector recovery into an augmented four-step problem. Simulation results reveal that the augmented approach achieves the best performance, while both proposed methods outperform the natural two-step technique with substantially less computational requirements.

In this paper, we will investigate the efficacy of IMAT (Iterative Method of Adaptive Thresholding) in recovering the sparse signal (parameters) for linear models with missing data. Sparse recovery rises in compressed sensing and machine learning problems and has various applications necessitating viable reconstruction methods specifically when we work with big data. This paper will focus on comparing the power of IMAT in reconstruction of the desired sparse signal with LASSO. Additionally, we will assume the model has random missing information. Missing data has been recently of interest in big data and machine learning problems since they appear in many cases including but not limited to medical imaging datasets, hospital datasets, and massive MIMO. The dominance of IMAT over the well-known LASSO will be taken into account in different scenarios. Simulations and numerical results are also provided to verify the arguments.