Tensor completion via multi-shared-modes canonical correlation analysis

In real-world traffic data, missing data is inevitable for traffic collection problem and some other reasons. Therefore, it is increasingly critical to complete the whole traffic data. To make full use of hidden spatial-temporal structures of the Internet traffic data, we extend to traffic matrix (user, server) to a three-way tensor (user, server, time). This paper proposed a novel algorithm to recover the missing data, named TSVT (Tucker decomposition and tensor singular value thresholding). We exploit the proposed the Tucker decomposition and low rank tensor completion (LRTC). Based on LRTC, we extend the known singular value thresholding (SVT) to the tensor case, and combine Tucker decomposition to optimize the performance of the algorithm. Based on Internet traffic data, we model two different tensors, and apply our TSVT to infer the unobserved data of the traffic. Furthermore, we apply TSVT algorithm to our Internet data. Numerical experiments on Synthetic data verify our method and the application for completing DPI traffic data proves the effectiveness of the method.

With the advance of acquisition techniques, plentiful higherorder tensor data sets are built up in a great variety of fields such as computer vision, neuroscience, remote sensing and recommender systems. The real-world tensors often contain missing values, which makes tensor completion become a prerequisite to utilize them. Previous studies have shown that imposing a low-rank constraint on tensor completion produces impressive performances. In this paper, we argue that low-rank constraint, albeit useful, is not effective enough to exploit the local smooth and piecewise priors of visual data. We propose integrating total variation into low-rank tensor completion (LRTC) to address the drawback. As LRTC can be formulated by both tensor unfolding and tensor decomposition, we develop correspondingly two methods, namely LRTC-TV-I and LRTC-TVII, and their iterative solvers. Extensive experimental results on color image and medical image inpainting tasks show the effectiveness and superiority of the two methods against state-of-the-art competitors.

Tensor Canonical Correlation Analysis (TCCA) is a commonly employed statistical method utilized to examine linear associations between two sets of tensor datasets. However, the existing TCCA models fail to adequately address the heterogeneity present in real-world tensor data, such as brain imaging data collected from diverse groups characterized by factors like sex and race. Consequently, these models may yield biased outcomes. In order to surmount this constraint, we propose a novel approach called Multi-Group TCCA (MG-TCCA), which enables the joint analysis of multiple subgroups. By incorporating a dual sparsity structure and a block coordinate ascent algorithm, our MG-TCCA method effectively addresses heterogeneity and leverages information across different groups to identify consistent signals. This novel approach facilitates the quantification of shared and individual structures, reduces data dimensionality, and enables visual exploration. To empirically validate our approach, we conduct a study focused on investigating correlations between two brain positron emission tomography (PET) modalities (AV-45 and FDG) within an Alzheimer's disease (AD) cohort. Our results demonstrate that MG-TCCA surpasses traditional TCCA in identifying sex-specific cross-modality imaging correlations. This heightened performance of MG-TCCA provides valuable insights for the characterization of multimodal imaging biomarkers in AD.

Laplacian Eigenmaps (LE) is a classical nonlinear manifold learning algorithm defined on Euclidean space. One contribution of this paper is to extend the LE algorithm on Grassmann manifold which is the set of linear subspaces. Another contribution is to integrate the label information of training subspaces and the geometric distribution information of all subspaces. A Semi-Supervised Laplacian Eigenmaps algorithm on Grassmann Manifold (SSLE-GM) is proposed. Experimental results on Cambridge gesture database showed that SSLE-GM obtained the highest recognition rate by comparing with Mutual Subspace Method (MSM), Constrained Mutual Subspace Method (CMSM), Tensor Canonical Correlation Analysis (TCCA), Canonical Correlation Analysis (CCA), and probabilistic Latent Semantic Analysis (pLSA).

Seismic data are often undersampled owing to physical or financial limitations. However, complete and regularly sampled data are becoming increasingly critical in seismic processing. In this paper, we present an efficient two-dimensional (2D) seismic data reconstruction method that works on texture-based patches. It performs completion on a patch tensor, which folds texture-based patches into a tensor. Reconstruction is performed by reducing the rank using tensor completion algorithms. This approach differs from past methods, which proceed by unfolding matrices into columns and then applying common matrix completion approaches to deal with 2D seismic data reconstruction. Here, we first re-arrange the seismic data matrix into a third-order patch tensor, by stacking texture-based patches that are divided from seismic data. Then, the seismic data reconstruction problem is formulated into a low-rank tensor completion problem. This formulation avoids destroying the spatial structure, and better extracts the underlying useful information. The proposed method is efficient and gives an improved performance compared with traditional approaches. The effectiveness of our patch tensor-based framework is validated using two classical tensor completion algorithms, low-rank tensor completion (LRTC), and the parallel matrix factorization algorithm (TMac), on both synthetic and field data experiments.

Low-rank tensor completion (LRTC) has gained significant attention due to its powerful capability of recovering missing entries. However, it has to repeatedly calculate the time-consuming singular value decomposition (SVD). To address this drawback, we, based on the tensor-tensor product (t-product), propose a new LRTC method-the unified tensor factorization (UTF)-for 3-way tensor completion. We first integrate the tensor factorization (TF) and the tensor nuclear norm (TNN) regularization into a framework that inherits the benefits of both TF and TNN: fast calculation and convex optimization. The conditions under which TF and TNN are equivalent are analyzed. Then, UTF for tensor completion is presented and an efficient iterative updated algorithm based on the alternate direction method of multipliers (ADMM) is used for our UTF optimization, and the solution of the proposed alternate minimization algorithm is also proven to be able to converge to a Karush–Kuhn–Tucker (KKT) point. Finally, numerical experiments on synthetic data completion and image/video inpainting tasks demonstrate the effectiveness of our method over other state-of-the-art tensor completion methods.

Traffic sensor networks allow convenient collection of travel data that are of great significance for intelligent transportation systems (ITSs). However, the universality of missing data impedes the application of ITS and thus accurate missing data recovery is indispensable in practice. Typically, the global low-rankness and local spatiotemporal smoothness exist in underlying traffic tensor data. In light of this, this article proposes an improved low-rank tensor completion (LRTC) model by exploiting abundant structural information from incomplete tensors. Specifically, a logarithm power composite (LPC)-norm is first proposed as a nonconvex substitute of the rank function, leading to a flexible characterization of tensor multidimensional correlation. Then, a joint structural regression (JSR) model is presented to simultaneously leverage the intrinsic temporal continuity and profile similarity of traffic data. By doing so, we construct a novel nonconvex LRTC model by integrating the global low-rankness and fine-grained spatiotemporal structure that are complementary to each other. To solve the proposed model, following the optimization framework of the alternating direction method of multipliers (ADMMs), we develop an efficient iterative algorithm where each step can be solved in a closed form. Extensive experiments on four real-world traffic data are conducted to evaluate the effectiveness of the proposed approach. The results demonstrate that compared with other tensor completion methods, our model significantly improves the recovery performance.

The presence of element failure results in an inevitable performance loss in angle estimation in multiple-input multiple-output (MIMO) radar. In this article, we consider the angle estimation problem for bistatic MIMO radar under element failure. To exploit the multidimensional structure, a covariance tensor of the uniform linear array (ULA)-based MIMO radar is constructed, where some slices are entirely missing due to faults in the array elements. Then, recovering failed-element signals can be formulated as a low-rank tensor completion (LRTC) problem with structurally missing entries. To address this problem, we propose a novel tensor completion approach via CANDECOMP/PARAFAC decomposition with factor priors. The essence of the proposed method is to fully exploit not only the Vandermonde structure of factor matrices but also their correlations. To enforce these factor priors, we formulate an optimization problem that consists of an objective function penalizing the nuclear norm of block Hankel matrices formed by the factor matrices and the constraints to reveal the relationship among the factor matrices. To solve the optimization problem, we develop an algorithm based on the alternating direction method of multipliers (ADMM), thereby recovering the signals of failed elements. Finally, conventional algorithms yield robust angle estimation. Simulation results verify the effectiveness of the proposed algorithm for dealing with element failure.

Two new low rank tensor completion methods based on sum nuclear norm

NT-DPTC: A non-negative temporal dimension preserved tensor completion model for missing traffic data imputation

Complete traffic sensor data is considered to be one of the critical ingredients for intelligent transportation systems (ITS). However, the traffic measurements prevalently suffer from the inevitable and ubiquitous missing values. Current missing data completion algorithms are difficult to leverage the global low-rank property and the fine-grained spatial-temporal structure simultaneously. To circumvent this problem, this work presents a novel collaborative nonconvex low-rank spatial-temporal data tensor completion model that can take full advantage of the inherent spatial-temporal characteristics of traffic measurement data. First, the tensor Schatten p-norm, as an effective nonconvex surrogate of rank function, is used to exploit the global multi-dimensional correlation of traffic data. Then, we present an elastic net self-representation method and utilize an autoregression model in order to simultaneously capture the self-similarity and the temporal continuity of traffic data acquired in the same sensor network. By integrating the above elements in a unified nonconvex learning model, our method can explore the inherent structure of traffic data from the viewpoints of both global multi-dimensional correlation and fine-grained spatial and temporal dependency. Then, in the view of the general framework of the alternating directions method of multipliers (ADMM), an efficient iterative algorithm is designed to solve our model. Besides, to optimize the parameter combination of the model, a Bayesian optimization-based parameter selection algorithm is developed, which avoids manual parameter adjustment. Extensive experiments and analyses on two real-world traffic datasets are carried out. The results demonstrate the advantages of our model under diverse missing patterns and missing ratios.

Low-rank (LR) and total variation (TV) are two most frequent priors that occur in image processing problems, and they have sparked a tremendous amount of researches, particularly for moving from scalar to vector, matrix or even high-order based functions. However, discretization schemes used for TV regularization often ignore the difference of the intrinsic properties, so it will lead to the problem that local smoothness cannot be effectively generated, let alone the problem of blurred edges. To address the image inpainting problem with corrupted data, in this paper, the color images are naturally considered as three-dimensional tensors, whose prior of smoothness can be measured by varietal TV norm along different dimensions. Specifically, we propose incorporating Shannon total variation (STV) and low-rank tensor completion (LRTC) into the construction of the final cost function, in which a new nonconvex low-rank constraint, namely truncated $\gamma $ -norm, is involved for closer rank approximation. Moreover, two methods are developed, i.e., LRRSTV and LRRSTV-T, due to the fact that LRTC can be represented by tensor unfolding and tensor decomposition. The final solution can be achieved by a practical variant of the augmented Lagrangian alternating direction method (ALADM). Experiments on color image inpainting tasks demonstrate that the proposed methods perform better then the state-of-the-art algorithms, both qualitatively and quantitatively.

Monitoring the performance of a large network would involve a high measurement cost. To reduce the overhead, sparse network monitoring techniques may be applied to select paths or time intervals to take the measurements, while the remaining monitoring data can be inferred leveraging the spatial-temporal correlations among data. The quality of missing data recovery, however, highly relies on the specific inference technique adopted. Tensor completion is a promising technique for more accurate missing data inference by exploiting the multi-dimensional data structure. However, data processing for higher dimensional tensors involves a large amount of computation, which prevents conventional tensor completion algorithms from practical application in the presence of large amount of data. This work takes the initiative to investigate the potential and methodologies of performing parallel processing for high-speed and high accuracy tensor completion over Graphics Processing Units (GPUs). We propose a GPU-accelerated parallel Tensor Completion scheme (GPU-TC) for accurate and fast recovery of missing data. To improve the data recovery accuracy and speed, we propose three novel techniques to well exploit the tensor factorization structure and the GPU features: grid-based tensor partition, independent task assignment based on Fisher-Yates shuffle, sphere facilitated and memory-correlated scheduling. We have conducted extensive experiments using network traffic trace data to compare the proposed GPU-TC with the state of art tensor completion algorithms and matrix-based algorithms. The experimental results demonstrate that GPU-TC can achieve significantly better performance in terms of two relative error ratio metrics and computation time.

Binocular stereo vision is a common way for 3D reconstruction of objects, which is core technology in the fields of computer-aided geometric design, machine vision and so on. However, the conventional binocular stereo vision algorithms are very sensitive to the ambient lighting, restricting their further promotion. The method of tensor completion, applied to the progress of preprocessing the captured binocular image, provides a way to solve this problem. In this paper, a binocular stereo vision algorithm that uses low-rank tensor completion to perform 3D reconstruction is presented. In the pictures, the positions where the ambient illumination are strong are regarded as information that cannot be observed, and the others are regarded as observable information. The unobservable information will be completed using the observable information based on tensor completion. The results show that the 3D reconstruction results obtained using tensor completion are better than those without tensor completion.

Fusion of low-rankness and smoothness under learnable nonlinear transformation for tensor completion