Low-rank Completion Research Articles

SOC Algorithm-Based Framework for Spectrum Occupancy Completion

This paper investigates the completion of continuous spectrum occupancy in space-frequency domain. Most of the existing studies adopt the traditional scheme, namely field strength completion followed by feature extraction. However, the existence of non-ideal sampling data challenges the recovery of spectrum occupancy array in accuracy. In this paper, we first develop a spectrum occupancy completion framework that implements feature extraction and spectrum occupancy completion sequentially, which reduces the impact of non-ideal data on completion. Then, to further improve the completion accuracy, a semantic-optional-based completion (SOC) algorithm is proposed, which divides the continuous spectrum occupancy data into discrete and multi-class spectrum semantic data according to the spectrum occupancy level. Specifically, the completion of spectrum occupancy is realized by specific semantic extraction, spectrum semantic completion and spectrum semantic recovery. Further, the real-world measured data are simulated to demonstrate the effectiveness of our proposed framework and SOC. The experimental results show that, compared with the low-rank completion algorithm and spatial interpolation algorithm, the proposed SOC has higher completion accuracy at more than 93% frequency points.

Color Image Restoration Using Sub-Image Based Low-Rank Tensor Completion.

Many restoration methods use the low-rank constraint of high-dimensional image signals to recover corrupted images. These signals are usually represented by tensors, which can maintain their inherent relevance. The image of this simple tensor presentation has a certain low-rank property, but does not have a strong low-rank property. In order to enhance the low-rank property, we propose a novel method called sub-image based low-rank tensor completion (SLRTC) for image restoration. We first sample a color image to obtain sub-images, and adopt these sub-images instead of the original single image to form a tensor. Then we conduct the mode permutation on this tensor. Next, we exploit the tensor nuclear norm defined based on the tensor-singular value decomposition (t-SVD) to build the low-rank completion model. Finally, we perform the tensor-singular value thresholding (t-SVT) based the standard alternating direction method of multipliers (ADMM) algorithm to solve the aforementioned model. Experimental results have shown that compared with the state-of-the-art tensor completion techniques, the proposed method can provide superior results in terms of objective and subjective assessment.

Jointly Low-Rank Tensor Completion for Estimating Missing Spatiotemporal Values in Logistics Systems

With the deepening of industry 4.0 paradigm in logistics systems, artificial intelligent has been widely used to improve the quality of logistics services. Considering that data collected in comprehensive logistics service system usually integrate multistage complex information such as traffic flow records and spatiotemporal trajectory, it is inevitable that the data are incomplete and partially missing due to equipment failures, communication interruptions, etc. As an effective spatiotemporal completion tool in logistics systems, low-rank tensor completion has aroused extensive research interest thanks to its excellent performance on data recovery. Although existing tensor completion methods effectively capture the complex associations/dependencies of multidimensional inputs, they fail to exploit the potential characteristics of spatiotemporal data, such as the periodicity. In this article, we propose a jointly low-rank tensor completion method for logistics data completion, which constructs multiple periodic subtensors by setting an appropriate time window, then performs jointly low-rank completion and imputation. In addition, we also provide an optimization algorithm based on Alternating Direction Method of Multiplier framework for the proposed problem. Experimental results on four logistics-related datasets have further demonstrated the promising performance of the proposed method compared with other state-of-the-art competitors. We believe that the proposed approach not only effectively maintains the advantages of classical completion methods, but also fully excavates the multidimensional correlation and hidden patterns behind records, and further provides a novel and effective strategy for data completion and imputation in logistics systems.

Calibrationless multi-slice Cartesian MRI via orthogonally alternating phase encoding direction and joint low-rank tensor completion.

We propose a multi-slice acquisition with orthogonally alternating phase encoding (PE) direction and subsequent joint calibrationless reconstruction for accelerated multiple individual 2D slices or multi-slice 2D Cartesian MRI. Specifically, multi-slice multi-channel data are first acquired with random or uniform PE undersampling while orthogonally alternating PE direction between adjacent slices. They are then jointly reconstructed through a recently developed low-rank multi-slice Hankel tensor completion (MS-HTC) approach. The proposed acquisition and reconstruction strategy was evaluated with human brain MR data. It effectively suppressed aliasing artifacts even at high acceleration factor, outperforming the existing MS-HTC approach, where PE direction is the same between adjacent slices. More importantly, the new strategy worked robustly with uniform undersampling or random undersampling without any consecutive central k-space lines. In summary, our proposed multi-slice MRI strategy exploits both coil sensitivity and image content similarities across adjacent slices. Orthogonally alternating PE direction among slices substantially facilitates the low-rank completion process and improves image reconstruction quality. This new strategy is applicable to uniform and random PE undersampling. It can be easily implemented in practice for Cartesian parallel imaging of multiple individual 2D slices without any coil sensitivity calibration.

Hybrid texture synthesis and interpolated structure image completion

The issue of image completion has been extensively developed for a few dozen years and many computational methods exist to tackle various image completion problems. When only one incomplete image is available, the missing regions are relatively large but their texture can be recovered from surrounding regions, then the texture synthesis methods are usually the most efficient. If an incomplete image contains a large number of missing pixels (e.g. 90%) or small missing regions uniformly distributed across it, low-rank approximation or interpolation-based methods seem to be the best choice. However, when the image of interest contains both types of incompleteness, none of the mentioned methods can be applied. In this study, we propose a hybrid computational method combining the radial basis function interpolation with the texture synthesis to tackle the mentioned problem. The experiments, carried out on various image completion problems, demonstrate that our hybrid strategy considerably outperforms the baseline and selected low-rank completion methods.

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.

A Smoothness Regularized Low-Rank Completion Method for Power Grid Middle Platform Data

Currently, the construction of power middle platform plays a vitally important role in the evolution of the smart grid. However, due to sensor failures or network delays, the sampled power middle platform data is often inevitably missing. To address this challenge, in this paper, we propose a smoothness regularized low-rank completion method for power middle platform missing data. Technically, the acquired middle platform data are formed to a time-series data matrix. Then, the low-rank matrix recovery model is applied to complete the missing data. Since the middle platform data is time-continuous, we adopt a total variation term to use this piece-wise smoothness. Finally, the proposed model is efficiently solved by the distributed alternating direction method of multipliers. Experimental evaluations on a real middle platform dataset validate the performance of the proposed method.

Removing Adversarial Noise via Low-Rank Completion of High-Sensitivity Points.

Deep neural networks are fragile under adversarial attacks. In this work, we propose to develop a new defense method based on image restoration to remove adversarial attack noise. Using the gradient information back-propagated over the network to the input image, we identify high-sensitivity keypoints which have significant contributions to the image classification performance. We then partition the image pixels into the two groups: high-sensitivity and low-sensitivity points. For low-sensitivity pixels, we use a total variation (TV) norm-based image smoothing method to remove adversarial attack noise. For those high-sensitivity keypoints, we develop a structure-preserving low-rank image completion method. Based on matrix analysis and optimization, we derive an iterative solution for this optimization problem. Our extensive experimental results on the CIFAR-10, SVHN, and Tiny-ImageNet datasets have demonstrated that our method significantly outperforms other defense methods which are based on image de-noising or restoration, especially under powerful adversarial attacks.

An efficient method for non-negative low-rank completion

In this article, we propose a new method for low-rank completion of a large sparse matrix, subject to non-negativity constraint. As a challenging prototype of this problem, we have in mind the well-known Netflix problem. Our method is based on the derivation of a constrained gradient system and its numerical integration. The methods we propose are based on the constrained minimization of a functional associated to the low-rank completion matrix having minimal distance to the given matrix. In the main 2-level method, the low-rank matrix is expressed in the form of the non-negative factorization X = eUVT, where the factors U and V are assumed to be normalized with unit Frobenius norm. In the inner level—for a given e—we minimize the functional; in the outer level, we tune the parameter e until we reach a solution. Numerical experiments on well-known large test matrices show the effectiveness of the method when compared with other algorithms available in the literature.

Image Completion with Hybrid Interpolation in Tensor Representation

The issue of image completion has been developed considerably over the last two decades, and many computational strategies have been proposed to fill-in missing regions in an incomplete image. When the incomplete image contains many small-sized irregular missing areas, a good alternative seems to be the matrix or tensor decomposition algorithms that yield low-rank approximations. However, this approach uses heuristic rank adaptation techniques, especially for images with many details. To tackle the obstacles of low-rank completion methods, we propose to model the incomplete images with overlapping blocks of Tucker decomposition representations where the factor matrices are determined by a hybrid version of the Gaussian radial basis function and polynomial interpolation. The experiments, carried out for various image completion and resolution up-scaling problems, demonstrate that our approach considerably outperforms the baseline and state-of-the-art low-rank completion methods.

Clipped Matrix Completion: A Remedy for Ceiling Effects

We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a low-rank matrix from various information deficits by using the principle of low-rank completion. However, the current theoretical guarantees for low-rank MC do not apply to clipped matrices, as the deficit depends on the underlying values. Therefore, the feasibility of clipped matrix completion (CMC) is not trivial. In this paper, we first provide a theoretical guarantee for the exact recovery of CMC by using a trace-norm minimization algorithm. Furthermore, we propose practical CMC algorithms by extending ordinary MC methods. Our extension is to use the squared hinge loss in place of the squared loss for reducing the penalty of overestimation on clipped entries. We also propose a novel regularization term tailored for CMC. It is a combination of two trace-norm terms, and we theoretically bound the recovery error under the regularization. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.

Multichannel Hankel Matrix Completion Through Nonconvex Optimization

This paper studies the multichannel missing data recovery problem when the measurements are generated by a dynamical system. A new model, termed multichannel low-rank Hankel matrices, is proposed to characterize the intrinsic low-dimensional structures in multichannel time series. The data recovery problem is formulated as a nonconvex optimization problem, and two fast algorithms (AM-FIHT and RAM-FIHT), both with linear convergence rates, are developed to recover the missing points with provable performance guarantees. The required number of observations is significantly reduced, compared with conventional low-rank completion methods. Our methods are verified through numerical experiments on synthetic data and recorded synchrophasor data in power systems.

A Gradient System for Low Rank Matrix Completion

In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.

Fast missing-data IAA with application to notched spectrum SAR

Recently, the spectral estimation method known as the iterative adaptive approach (IAA) has been shown to provide higher resolution and lower sidelobes than comparable spectral estimation methods. The computational complexity is higher than methods such as the periodogram (matched filter method). Fast algorithms have been developed that considerably reduce the computational complexity of IAA by using Toeplitz and Vandermonde structures. For the missing-data case, several of these structures are lost, and existing fast algorithms are only efficient when the number of available samples is small. In this work, we consider the case in which the number of missing samples is small. This allows us to use low-rank completion to transform the problem to the structured problem. We compare the computational speed of the algorithm with the state of the art and demonstrate the utility in a frequency-notched synthetic aperture radar imaging problem.