Existing Matrix Completion Methods Research Articles

Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure

Array sensor failure poses a serious challenge to robust direction-of-arrival (DOA) estimation in complicated environments. Although existing matrix completion methods can successfully recover the damaged signals of an impaired sensor array, they cannot preserve the multi-way signal characteristics as the dimension of arrays expands. In this paper, we propose a structural-missing tensor completion algorithm for robust DOA estimation with uniform rectangular array (URA), which exhibits a high robustness to non-ideal sensor failure conditions. Specifically, the signals received at the impaired URA are represented as a three-dimensional incomplete tensor, which contains whole fibers or slices of missing elements. Due to this structural-missing pattern, the conventional low-rank tensor completion becomes ineffective. To resolve this issue, a spatio-temporal dimension augmentation method is developed to transform the structural-missing tensor signal into a six-dimensional Hankel tensor with dispersed missing elements. The augmented Hankel tensor can then be completed with a low-rank regularization by solving a Hankel tensor nuclear norm minimization problem. As such, the inverse Hankelization on the completed Hankel tensor recovers the tensor signal of an unimpaired URA. Accordingly, a completed covariance tensor can be derived and decomposed for robust DOA estimation. Simulation results verify the effectiveness of the proposed algorithm.

Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning.

The existing matrix completion methods focus on optimizing the relaxation of rank function such as nuclear norm, Schatten- p norm, etc. They usually need many iterations to converge. Moreover, only the low-rank property of matrices is utilized in most existing models and several methods that incorporate other knowledge are quite time-consuming in practice. To address these issues, we propose a novel non-convex surrogate that can be optimized by closed-form solutions, such that it empirically converges within dozens of iterations. Besides, the optimization is parameter-free and the convergence is proved. Compared with the relaxation of rank, the surrogate is motivated by optimizing an upper-bound of rank. We theoretically validate that it is equivalent to the existing matrix completion models. Besides the low-rank assumption, we intend to exploit the column-wise correlation for matrix completion, and thus an adaptive correlation learning, which is scaling-invariant, is developed. More importantly, after incorporating the correlation learning, the model can be still solved by closed-form solutions such that it still converges fast. Experiments show the effectiveness of the non-convex surrogate and adaptive correlation learning.

Matrix Completion Based on Low-Rank and Local Features Applied to Images Recovery and Recommendation Systems

Recent advances have shown that the challenging problem of matrix completion arises from real-world applications, such as image recovery, and recommendation systems. Existing matrix completion methods utilize the low-rank property of sparse matrices to fill missing entries, which essentially exploits the row-rank relationship and ignores the local features of sparse matrices. In this paper, we propose a novel matrix completion method that takes the path of combining local features and low-rank information. First, the original sparse matrix is processed so that the neighboring rows and columns of the sparse matrix are most similar to each other, and then the missing data are filled by the iterative rank-one matrix completion method. Moreover, we use two-dimensional convolutional operations of sparse matrices to obtain the local features to dredge up the missing entries. Finally, the two filled results are integrated to obtain the final missing entries. We conduct extensive experiments on the real-world datasets. The experimental results demonstrate the significant outperforms of the proposed method on five image datasets of different sizes, and show a strong competitive advantage on the recommender system datasets.

Iterative rank-one matrix completion via singular value decomposition and nuclear norm regularization

Matrix completion is widely used in many fields. In existing matrix completion methods, such as rank minimization and matrix factorization, the hyperparameters must be learned. However, hyperparameter tuning is a time-consuming and tedious process. In this paper, we propose a novel matrix completion method called IMC, i.e., iterative rank-one matrix completion via singular value decomposition (SVD) and nuclear norm regularization. First, we construct a rank-one matrix completion model using nuclear norm regularization. Then, the variables to be optimized in the model are divided into several blocks. Finally, the blocks are iteratively optimized one by one until convergence. For the optimization of each block, we propose an efficient solution scheme based on SVD, in which only the maximum singular value and leading singular vectors of a sparse matrix must be calculated. Further, a nonparametric singular value penalty function is designed to ensure a low-rank completion matrix. In addition, the optimization of each block uses only the values inside the observed entries; hence, no errors are accumulated. Test results shows that the proposed method converges rapidly and outperforms some state-of-the-art methods when applied to grayscale image restoration, recommendation systems, and vote networks.

Matrix Completion under Low-Rank Missing Mechanism

Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.

Collaborating filtering using unsupervised learning for image reconstruction from missing data

In the image acquisition process, important information in an image can be lost due to noise, occlusion, or even faulty image sensors. Therefore, we often have images with missing and/or corrupted pixels. In this work, we address the problem of image completion using a matrix completion approach that minimizes the nuclear norm to recover missing pixels in the image. The image matrix has a low rank. The proposed approach uses the nuclear norm function as a surrogate of the rank function in the aim to resolve the problem of rank minimization that is known as an NP-hard problem. It is an adaptation of the collaborating filtering approach used for users’ profile construction. The main advantage of this approach is that it uses a learning process to classify pixels into clusters and exploits them to run a predictive method in the aim to recover the missing or unknown data. For performance evaluation, the proposed approach and the existing matrix completion methods are compared for image reconstruction according to the PSNR measure. These methods are applied on a dataset composed of standard images used for image processing. All the recovered images obtained during experimentation are also dressed to compare them visually. Simulation results verify that the proposed approach achieves better performances than the existing matrix completion methods used for image reconstruction from missing data.

Enhancing Matrix Completion Using a Modified Second-Order Total Variation

In this paper, we propose a new method to deal with the matrix completion problem. Different from most existing matrix completion methods that only pursue the low rank of underlying matrices, the proposed method simultaneously optimizes their low rank and smoothness such that they mutually help each other and hence yield a better performance. In particular, the proposed method becomes very competitive with the introduction of a modified second-order total variation, even when it is compared with some recently emerged matrix completion methods that also combine the low rank and smoothness priors of matrices together. An efficient algorithm is developed to solve the induced optimization problem. The extensive experiments further confirm the superior performance of the proposed method over many state-of-the-art methods.

Accurate traffic matrix completion based on multi-Gaussian models

Traffic matrix (TM) describes the volumes of traffic between a set of sources and destinations in a network. As an important parameter, TMs are used in a variety of network engineering tasks, such as traffic engineering, capacity planning and anomaly detection. However, it is a challenge to reliably measure TMs in practice. For example, due to flaws in the measurement systems and possible failure in data collection systems, missing values are unavoidable. It is important to recover the missing data from the partial direct measurements. Existing matrix completion methods do not fully consider network traffic behavior and traffic hidden characteristic. Their completion accuracy tends to be significantly worse when the data loss rate is high. In this paper, we perform a study on intrinsic characteristics of network traffic by analyzing real-world traffic trace data, which reveals that traffic has the features of temporal stability and spatial affinity. According to traffic spatial feature, we model TM as multi-Gaussian distributions, which describes the actual network traffic more accurately. Furthermore, we propose a novel matrix completion method based on multi-Gaussian models to estimate the missing traffic data. Finally, we utilize traffic temporal characteristic to further optimize traffic matrix completion for the missing data interpolation. Our proposed approach has been evaluated utilizing real-world traffic trace data. The extensive experiments demonstrate that our method achieves significantly better performance compared with the state-of-the-art interpolation methods.

Low-Rank Matrix Completion in the Presence of High Coherence

Prevalent matrix completion methods capture only the low-rank property which gives merely a constraint that the data points lie on some low-dimensional subspace, but generally ignore the extra structures (beyond low-rank) that specify in more detail how the data points lie on the subspace. Whenever the data points are not uniformly distributed on the low-dimensional subspace, the row-coherence of the target matrix to recover could be considerably high and, accordingly, prevalent methods might fail even if the target matrix is fairly low-rank. To relieve this challenge, we suggest to consider a model termed low-rank factor decomposition (LRFD), which imposes an additional restriction that the data points must be represented as linear, compressive combinations of the bases in a given dictionary. We show that LRFD can effectively mitigate the challenges of high row-coherence, provided that its dictionary is configured properly. Namely, it is mathematically proven that if the dictionary is well-conditioned and low-rank, then LRFD can weaken the dependence on the row-coherence. In particular, if the dictionary itself is low-rank, then the dependence on the row-coherence can be entirely removed. Subsequently, we devise two practical algorithms to obtain proper dictionaries in unsupervised environments: one uses the existing matrix completion methods to construct the dictionary in LRFD, and the other tries to learn a proper dictionary from the data given. Experiments on randomly generated matrices and motion datasets show superior performance of our proposed algorithms.

Data reconstruction in internet traffic matrix

Traffic matrix is an abstract representation of the traffic volume flowing between sets of source and destination pairs. It is a key input parameter of network operations management, planning, provisioning and traffic engineering. Traffic matrix is also important in the context of OpenFlow-based networks. Because even good measurement systems can suffer from errors and data collection systems can fail, missing values are common. Existing matrix completion methods do not consider traffic exhibit characteristics and only provide a finite precision. To address this problem, this paper proposes a novel approach based on compressive sensing and traffic self-similarity to reconstruct the missing traffic flow data. Firstly, we analyze the real-world traffic matrix, which all exhibit low-rank structure, temporal smoothness feature and spatial self-similarity. Then, we propose Self-Similarity and Temporal Compressive Sensing (SSTCS) algorithm to reconstruct the missing traffic data. The extensive experiments with the real-world traffic matrix show that our proposed SSTCS can significantly reduce data reconstruction errors and achieve satisfactory accuracy comparing with the existing solutions. Typically SSTCS can successfully reconstruct the traffic matrix with less than 32% errors when as much as 98% of the data is missing.

Decomposition Approach for Low-Rank Matrix Completion and Its Applications

In this paper, we describe a low-rank matrix completion method based on matrix decomposition. An incomplete matrix is decomposed into sub-matrices which are filled with a proposed trimming step and then are recombined to form a low-rank completed matrix. The divide-and-conquer approach can significantly reduce computation complexity and storage requirement. Moreover, the proposed decomposition method can be naturally incorporated into any existing matrix completion methods to attain further gain. Unlike most existing approaches, the proposed method is not based on norm minimization nor on SVD decomposition. This makes it possible to be applied beyond real domain and can be used in arbitrary fields, including finite fields. The effectiveness of our proposed method is demonstrated through extensive numerical results on randomly generated and real matrix completion problems and a concrete application-video denoising. The numerical experiments show that the algorithm can reliably solve a wide range of problems at a speed significantly faster than recent algorithms. In the proposed denoising approach, we present a patch-based video denoising algorithm by grouping similar patches and then formulating the problem of removing noise using a decomposition approach for low-rank matrix completion. Experiments show that the proposed approach robustly removes mixed noise such as impulsive noise, Poisson noise, and Gaussian noise from any natural noisy video. Moreover, our approach outperforms state-of-the-art denoising techniques such as VBM3D and 3DWTF in terms of both time and quality. Our technique also achieves significant improvement over time against other matrix completion methods.

Robust Discrete Matrix Completion

Most existing matrix completion methods seek the matrix global structure in the real number domain and produce predictions that are inappropriate for applications retaining discrete structure, where an additional step is required to post-process prediction results with either heuristic threshold parameters or complicated mappings. Such an ad-hoc process is inefficient and impractical. In this paper, we propose a novel robust discrete matrix completion algorithm that produces the prediction from the collection of user specified discrete values by introducing a new discrete constraint to the matrix completion model. Our method achieves a high prediction accuracy, very close to the most optimal value of competitive methods with threshold values tuning. We solve the difficult integer programming problem via incorporating augmented Lagrangian method in an elegant way, which greatly accelerates the converge process of our method and provides the asymptotic convergence in theory. The proposed discrete matrix completion model is applied to solve three real-world applications, and all empirical results demonstrate the effectiveness of our method.