Low Permutation-Rank Matrices: Structural Properties and Noisy Completion

Trust networks, where people leave trust and distrust feedback, are becoming increasingly common. These networks may be regarded as signed graphs, where a positive edge weight captures the degree of trust while a negative edge weight captures the degree of distrust. Analysis of such signed networks has become an increasingly important research topic. One important analysis task is that of sign inference, i.e., infer unknown (or future) trust or distrust relationships given a partially observed signed network. Most state-of-the-art approaches consider the notion of structural balance in signed networks, building inference algorithms based on information about links, triads, and cycles in the network. In this paper, we first show that the notion of weak structural balance in signed networks naturally leads to a global low-rank model for the network. Under such a model, the sign inference problem can be formulated as a low-rank matrix completion problem. We show that we can perfectly recover missing relationships, under certain conditions, using state-of-the-art matrix completion algorithms. We also propose the use of a low-rank matrix factorization approach with generalized loss functions as a practical method for sign inference - this approach yields high accuracy while being scalable to large signed networks, for instance, we show that this analysis can be performed on a synthetic graph with 1.1 million nodes and 120 million edges in 10 minutes. We further show that the low-rank model can be used for other analysis tasks on signed networks, such as user segmentation through signed graph clustering, with theoretical guarantees. Experiments on synthetic as well as real data show that our low rank model substantially improves accuracy of sign inference as well as clustering. As an example, on the largest real dataset available to us (Epinions data with 130K nodes and 840K edges), our matrix factorization approach yields 94.6% accuracy on the sign inference task as compared to 90.8% accuracy using a state-of-the-art cycle-based method - moreover, our method runs in 40 seconds as compared to 10,000 seconds for the cycle-based method.

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.

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 considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as [Formula: see text] norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the [Formula: see text]-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this [Formula: see text] norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated [Formula: see text] norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated [Formula: see text] norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.

Compressive sensing enables efficient acquisition of hyperspectral images (HSIs) by assuming high redundancy on natural scenes. Several reconstruction algorithms have been proposed to retrieve the underlying image, and most of them take advantage of the spatial and spectral correlations. However, reconstruction may not be necessary in certain applications such as land cover classification. Instead of knowing the full image, researchers are interested in features that could be extracted directly from the compressed measurements, which provide high inference capabilities. Low-rank (LR) matrix approximation has been widely used in feature extraction (FE), because it reduces the data dimension and computational cost. Therefore, in this paper, compressive hyperspectral imaging and FE are combined in a framework for HSI classification using an LR matrix approximation model. In the proposed framework, the compressed measurements are acquired from a single-pixel spectrometer. Instead of using the traditional high-complexity reconstruction model, an LR matrix factorization problem is formulated. The LR problem maximizes the posterior distribution with respect to the feature space and coefficients, and it is numerically solved based on an alternating optimization strategy. By incorporating spatial information, the numerical procedure minimizes the total variational of the feature coefficients subject to an orthogonality constraint for the feature space. Experiments on real HSIs show that the proposed approach can provide equally competitive classification results when compared to the traditional approach that performs FE and classification on the recovered images.

The problem of matrix approximation appears ubiquitously in recommendation systems, computer vision and text mining. The prevailing assumption is that the partially observed matrix has a low-rank or can be well approximated by a low-rank matrix. However, this assumption is strictly that the partially observed matrix is globally low rank. In this paper, we propose a local sensitive formulation of matrix approximation which relaxes the global low-rank assumption, leading to a representation of the observed matrix as a weighted sum of low-rank matrices. We solve the problem by nonconvex optimization which exhibits superior performance of low rank matrix estimation when compared with convex relaxation. Our experiments show improvements in prediction accuracy over classical approaches for recommendation tasks.

In recent years, low rank matrix theory becomes one of the research hotspots in the image processing field. In this report, we construct a linear equation via born approximation for the electromagnetic inverse scattering problem, then solve the linear equation with Low Rank Constraint. The low rank matrix representation is applied to quickly and precisely reconstruct the low rank data matrix in GPR imaging. Simulation results and performance analysis show that Low Rank Reconstruction helps to reduce the memory space and computational complexity in GPR imaging.

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.

Image restoration via wavelet-based low-rank tensor regularization

Signal modeling refers to representing the signal via the parameters of some model. For example, signals are modeled as sums of sinusoids, estimating the parameters of these sinusoids, and storing the parameters instead of the original samples. The application of signal modeling is signal compression, prediction, reconstruction, feature extraction, information extraction, and recognition and identification. Some standard signal modelings are ARMA, MA, AR. There is also low rank signal modeling which represents the signal samples through low rank matrices. These are used in computer vision, data mining, and bioinformatics. In low rank modeling, the matrix completion and dimensionality reduction is important. In image compression, the JPEG coding can be mentioned as a well known application of signal modeling. Adaptive filters, and their applications, parametric, and non-parametric signal are discussed in this chapter.

We recently proposed an accelerated dynamic magnetic resonance imaging (MRI) reconstruction algorithm that exploits the underlying low rank and sparse properties of the data to achieve highly accelerated reconstructions. In this paper, we validate our algorithm in the context of dynamic free breathing cardiac Perfusion MRI on the Physiologically Improved Non Uniform Cardiac Torso Phantom, PINCAT phantom. The practical utilities of our scheme in providing significantly better reconstructions at higher accelerations in comparison to existing methods are studied. We demonstrate that our scheme do not have trade offs with accurate temporal modeling and spatial quality unlike the existing low rank based schemes. Our results also show the capability of our scheme to achieve better reconstruction qualities at high accelerations in comparison to using only the low rank or sparsity properties individually. We argue that the speed up obtained by our scheme could be capitalized in perfusion imaging to provide better spatio-temporal resolutions and volume coverage while the subject is freely breathing.

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.

To solve the problem that both training and test samples are corrupted due to occlusion and disguise during face recognition, a new method which is based on low rank matrix recovery with structural incoherence and low rank projection (LRSI_LRP) is presented. First, the training images are decomposed into a set of clean images and sparse errors via LRSI, and the derived clean images from distinct classes are forced to be as independent as possible by introducing structural incoherence regularization term into robust PCA. Then a low-rank projection matrix is learned based on the original training images and the recovered clean ones, and this low-rank projection matrix can correct corrupted test samples by projecting them onto their corresponding underlying subspaces. Finally, the corrected test samples are classified based on sparse representation-based classification (SRC). Experimental results on AR and Extended Yale B databases verify the efficacy and robustness of the proposed method.

Hyperspectral image (HSI) denoising is an effective image processing technology to improve the quality of HSI. By exploiting the spectral-spatial characteristics of HSI, this paper proposes a HSI denoising method based on the Graph-structured Low Rank model and Non-Local constraint (GLRNL). Graph-structured low rank model can reveal the spectral similarity in HSI and preserve the distribution characteristics of spectral bands. The graph structure is used to ensure that the denoised HSI has similar spectral characteristics as the real HSI, and reduce the spectral distortions. Moreover, the non-local constraint in GLRNL adopts the spatial-spectral self-similarity and provides an effective side prior information for low rank calculation. Comparing with some traditional low rank and sparse model based HSI denoising methods, the proposed method achieves the competitive results on different simulated and real HSI data with various types of noise.

We consider the problem of reconstructing a low rank matrix from noisy observations of a subset of its entries. This task has applications in statistical learning, computer vision, and signal processing. In these contexts, `noise' generically refers to any contribution to the data that is not captured by the low-rank model. In most applications, the noise level is large compared to the underlying signal and it is important to avoid overfitting. In order to tackle this problem, we define a regularized cost function well suited for spectral reconstruction methods. Within a random noise model, and in the large system limit, we prove that the resulting accuracy undergoes a phase transition depending on the noise level and on the fraction of observed entries. The cost function can be minimized using OPTSPACE (a manifold gradient descent algorithm). Numerical simulations show that this approach is competitive with state-of-the-art alternatives.