Colorization by Matrix Completion

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problem of matrix completion and robust principal component analysis, namely recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries and from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, we consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the -norm in this paper. Meanwhile, we put forward a novel and effective algorithm called augmented Lagrange multipliers to exactly solve the problem. For mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy image following an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality performance of our method is dominant if images possess low-rank features such as geometrically regular textures and similar structured contents; especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and the restoration ability for a low-rank image matrix, but also in the preservation of textures and details in the image.

The essence of distantly supervised relation extraction is that it is an incomplete multi-label classification problem with sparse and noisy features. To tackle the sparsity and noise challenges, we propose solving the classification problem using matrix completion on factorized matrix of minimized rank. We formulate relation classification as completing the unknown labels of testing items (entity pairs) in a sparse matrix that concatenates training and testing textual features with training labels. Our algorithmic framework is based on the assumption that the rank of item-byfeature and item-by-label joint matrix is low. We apply two optimization models to recover the underlying low-rank matrix leveraging the sparsity of feature-label matrix. The matrix completion problem is then solved by the fixed point continuation (FPC) algorithm, which can find the global optimum. Experiments on two widely used datasets with different dimensions of textual features demonstrate that our low-rank matrix completion approach significantly outperforms the baseline and the state-of-the-art methods.

Recovering a low-rank matrix from some of its linear measurements is a popular problem in many areas of science and engineering. One special case of it is the matrix completion problem, where we need to reconstruct a low-rank matrix from incomplete samples of its entries. A lot of efficient algorithms have been proposed to solve this problem and they perform well when Gaussian noise with a small variance is added to the given data. But they can not deal with the sparse random-valued noise in the measurements. In this paper, we propose a robust method for recovering the low-rank matrix with adaptive outlier pursuit when part of the measurements are damaged by outliers. This method will detect the positions where the data is completely ruined and recover the matrix using correct measurements. Numerical experiments show the accuracy of noise detection and high performance of matrix completion for our algorithms compared with other algorithms.

formula omitted] completions on partial matrices

Given the superposition of a low-rank matrix plus the product of a known fat\ncompression matrix times a sparse matrix, the goal of this paper is to\nestablish deterministic conditions under which exact recovery of the low-rank\nand sparse components becomes possible. This fundamental identifiability issue\narises with traffic anomaly detection in backbone networks, and subsumes\ncompressed sensing as well as the timely low-rank plus sparse matrix recovery\ntasks encountered in matrix decomposition problems. Leveraging the ability of\n$\\ell_1$- and nuclear norms to recover sparse and low-rank matrices, a convex\nprogram is formulated to estimate the unknowns. Analysis and simulations\nconfirm that the said convex program can recover the unknowns for sufficiently\nlow-rank and sparse enough components, along with a compression matrix\npossessing an isometry property when restricted to operate on sparse vectors.\nWhen the low-rank, sparse, and compression matrices are drawn from certain\nrandom ensembles, it is established that exact recovery is possible with high\nprobability. First-order algorithms are developed to solve the nonsmooth convex\noptimization problem with provable iteration complexity guarantees. Insightful\ntests with synthetic and real network data corroborate the effectiveness of the\nnovel approach in unveiling traffic anomalies across flows and time, and its\nability to outperform existing alternatives.\n

We propose a low rank structured matrix completion algorithm for image inpainting problems originated from scanning microscopy. The proposed method exploits the annihilation property observed in Gaussian Markov Random Field (GMRF) or partial differential equation (PDE)-based inpainting approaches. By utilizing the commutative property of the convolution, the annihilation property is embodied into rank-deficient block Hankel structure data matrices and the image inpainting problem is converted into low-rank structured matrix completion problem. To solve the structured low-rank matrix completion problem, an alternating direction method of multiplier (ADMM) method is used with factorization matrix initialization using the low rank matrix fitting (LMaFit) algorithm. Experimental results showed that the proposed method outperforms the existing state-of-the-art image inpainting methods.

This paper presents a low-rank matrix completion approach to tackle the problem of wall clutter mitigation for through-wall radar imaging in the compressive sensing context. In particular, the task of wall clutter removal is reformulated as a matrix completion problem in which a low-rank matrix containing wall clutter is reconstructed from compressive measurements. The proposed model regularizes the low-rank prior of the wall-clutter matrix via the nuclear norm, casting the wall-clutter mitigation task as a nuclear-norm penalized least squares problem. To solve this optimization problem, an iterative algorithm based on proximal gradient technique is introduced. Experiments on simulated full-wave electromagnetic data are conducted under compressive sensing scenarios. The results show that the proposed matrix completion approach is very effective at suppressing unwanted wall clutter and enhancing the desired targets.

This paper presents a low-rank matrix completion approach to tackle the problem of wall clutter mitigation for through-wall radar imaging in the compressive sensing context. In particular, the task of wall clutter removal is reformulated as a matrix completion problem in which a low-rank matrix containing wall clutter is reconstructed from compressive measurements. The proposed model regularizes the low-rank prior of the wall-clutter matrix via the nuclear norm, casting the wall-clutter mitigation task as a nuclear-norm penalized least squares problem. To solve this optimization problem, an iterative algorithm based on the proximal gradient technique is introduced. Experiments on simulated full-wave electromagnetic data are conducted under compressive sensing scenarios. The results show that the proposed matrix completion approach is very effective at suppressing unwanted wall clutter and enhancing the targets.

Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability task subsumes compressed sensing and the timely low-rank plus sparse matrix recovery encountered in matrix decomposition problems. Leveraging the ability of l 1 - and nuclear norms to recover sparse and low-rank matrices, a convex program is formulated to estimate the unknowns. Analysis and simulations confirm that the said convex program can recover the unknowns for sufficiently low-rank and sparse enough components, along with a compression matrix possessing an isometry property.

This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample complexity bound that is order-wise optimal with respect to the incoherence parameter (as well as to the rank $r$ and the matrix dimension $n$ up to a log factor). As a consequence, we improve the sample complexity of recovering a semidefinite matrix from $O(nr^{2}\log^{2}n)$ to $O(nr\log^{2}n)$, and the highest allowable rank from $\Theta(\sqrt{n}/\log n)$ to $\Theta(n/\log^{2}n)$. The key step in proof is to obtain new bounds on the $\ell_{\infty,2}$-norm, defined as the maximum of the row and column norms of a matrix. To illustrate the applicability of our techniques, we discuss extensions to SVD projection, structured matrix completion and semi-supervised clustering, for which we provide order-wise improvements over existing results. Finally, we turn to the closely-related problem of low-rank-plus-sparse matrix decomposition. We show that the joint incoherence condition is unavoidable here for polynomial-time algorithms conditioned on the Planted Clique conjecture. This means it is intractable in general to separate a rank-$\omega(\sqrt{n})$ positive semidefinite matrix and a sparse matrix. Interestingly, our results show that the standard and joint incoherence conditions are associated respectively with the information (statistical) and computational aspects of the matrix decomposition problem.

Data-driven approaches are widely applied for fault detection in industrial process. Recently, a new method for fault detection called principal component pursuit(PCP) is introduced. PCP is not only robust to outliers, but also can accomplish the objectives of model building, fault detection, fault isolation and process reconstruction simultaneously. PCP divides the data matrix into two parts: a fault-free low rank matrix and a sparse matrix with sensor noise and process fault. The statistics presented in this paper fully utilize the information in data matrix. Since the low rank matrix in PCP is similar to principal components matrix in PCA, a T2 statistic is proposed for fault detection in low rank matrix. And this statistic can illustrate that PCP is more sensitive to small variations in variables than PCA. In addition, in sparse matrix, a new monitored statistic performing the online fault detection with PCP-based method is introduced. This statistic uses the mean and the correlation coefficient of variables. Monte Carlo simulation and Tennessee Eastman (TE) benchmark process are provided to illustrate the effectiveness of monitored statistics.

The development of matrix completion theory provides new approaches for data gathering in Wireless Sensor Networks (WSN). The existing matrix completion algorithms for WSN mainly consider how to reduce the sampling number without considering the real-time performance when recovering the data matrix. In order to guarantee the recovery accuracy and reduce the recovery time consumed simultaneously, we propose a new Augmented Lagrange Multiplier (ALM) algorithm to recover the weather monitoring data. A large amount of experiments have been carried out to investigate the performance of the proposed ALM algorithm by using different parameter settings, different sampling rates and sampling models. In addition, we compare the proposed ALM algorithm with some existing algorithms in the literature. Experimental results show that the ALM algorithm can obtain better overall recovery accuracy with less computing time, which demonstrate that the ALM algorithm is an effective and efficient approach for recovering the real world weather monitoring data in WSN.

The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of interest. This model corresponds, for example, to ratings in recommender systems or labels in multi-class classification. We consider a general, non-uniform, sampling scheme and give theoretical guarantees on the performance of a constrained, nuclear norm penalized maximum likelihood estimator. One important advantage of this estimator is that it does not require knowledge of the rank or an upper bound on the nuclear norm of the unknown matrix and, thus, it is adaptive. We provide lower bounds showing that our estimator is minimax optimal. An efficient algorithm based on lifted coordinate gradient descent is proposed to compute the estimator. A limited Monte-Carlo experiment, using both simulated and real data is provided to support our claims.

The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical nonlinear conjugate gradients, developed within the framework of retraction-based optimization on manifolds. We describe all the necessary objects from differential geometry necessary to perform optimization over this low-rank matrix manifold, seen as a submanifold embedded in the space of matrices. In particular, we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions. Finally, we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations. The numerical experiments indicate that our approach scales very well for large-scale problems and compares favorably with the state-of-the-art, while outperforming most existing solvers.

Since lots of miRNA-disease associations have been verified, it is meaningful to discover more miRNA-disease associations for serving disease diagnosis and prevention of human complex diseases. However, it is not practical to identify potential associations using traditional biological experimental methods since the process is expensive and time consuming. Therefore, it is necessary to develop efficient computational methods to accomplish this task. In this work, we introduced a matrix completion model with dual Laplacian regularization (DLRMC) to infer unknown miRNA-disease associations in heterogeneous omics data. Specifically, DLRMC transformed the task of miRNA-disease association prediction into a matrix completion problem, in which the potential missing entries of the miRNA-disease association matrix were calculated, the missing association can be obtained based on the prediction scores after the completion procedure. Meanwhile, the miRNA functional similarity and the disease semantic similarity were fully exploited to serve the miRNA-disease association matrix completion by using a dual Laplacian regularization term. In the experiments, we conducted global and local Leave-One-Out Cross Validation (LOOCV) and case studies to evaluate the efficacy of DLRMC on the Human miRNA-disease associations dataset obtained from the HMDDv2.0 database. As a result, the AUCs of DLRMC is 0.9174 and 0.8289 in global LOOCV and local LOOCV, respectively, which significantly outperform a variety of previous methods. In addition, in the case studies on four significant diseases related to human health including Colon Neoplasms, Kidney neoplasms, Lymphoma and Prostate neoplasms, 90%, 92%, 92% and 94% out of the top 50 predicted miRNAs has been confirmed, respectively.