Robust Tensor Recovery Research Articles

Robust tensor recovery via a nonconvex approach with ket augmentation and auto‐weighted strategy

AbstractIn this article, we introduce a nonconvex tensor recovery approach, which employs the powerful ket augmentation technique to expand a low order tensor into a high‐order one so that we can exploit the advantage of tensor train (TT) decomposition tailored for high‐order tensors. Moreover, we define a new nonconvex surrogate function to approximate the tensor rank, and develop an auto‐weighted mechanism to adjust the weights of the resulting high‐order tensor's TT ranks. To make our approach robust, we add two mode‐unfolding regularization terms to enhance the model for the purpose of exploring spatio‐temporal continuity and self‐similarity of the underlying tensors. Also, we propose an implementable algorithm to solve the proposed optimization model in the sense that each subproblem enjoys a closed‐form solution. A series of numerical results demonstrate that our approach works well on recovering color images and videos.

Robust low tubal rank tensor recovery via [formula omitted]E criterion

Tensor analysis has received enormous attention as the increasing prevalence of high-dimensional data in science research and engineering application. Tensor recovery is an important and meaningful problem for tensor analysis. It aims to complete a tensor from an observed subset of its entries disturbed by noise. However, classical methods either develop on the second-order statistics or Lasso-type penalty, leading to them not effectively dealing with gross or dense noise effectively. To address such issues, we propose a robust tensor recovery model for simultaneously completing a low tubal rank tensor with complex noise and missing data. Based on tensor–tensor product (t-product), we first develop a tensor factor Frobenius norm to exploit the low tubal rank property which is closely related to tensor nuclear norm and tubal rank. By utilizing the robust L2 criterion, we derive the nonconvex objective function to accommodate the low tubal rank tensor. An implementable alternating minimization algorithm has been developed to estimate the low tubal rank tensor. It is worth noting that our method is able to jointly estimate the precision parameter to capture the hidden complex noise pattern. Furthermore, some convergence properties of the proposed algorithm are presented. A series of numerical experiments are conducted on both synthetic and real-world data to demonstrate the effectiveness and robustness of the proposed approach in comparison with the state-of-the-art methods.

Robust tensor recovery with nonconvex and nonsmooth regularization

This paper considers the least squares loss minimization problem regularized by tensor average rank and zero norm, to decompose the noisy observation into a low-rank tensor, a sparse tensor and a noise tensor. Due to the nonconvex and nonsmooth of the average rank and the zero norm of tensors, it is not easy to solve the problem directly. We construct the variational characterizations of the tensor average rank and the tensor zero norm, get an equivalent mathematical program with an equilibrium constraint, and then propose an equivalent Lipschitz surrogate for the regularization problem by adding the equality constraint into the objective. Moreover, we design a convex algorithm with multi-stage to verify the efficiency of the proposed method. Numerical experiments are reported for simulation data and actual data to show the performance of the new method.

Streaming Data Preprocessing via Online Tensor Recovery for Large Environmental Sensor Networks

Measuring the built and natural environment at a fine-grained scale is now possible with low-cost urban environmental sensor networks. However, fine-grained city-scale data analysis is complicated by tedious data cleaning including removing outliers and imputing missing data. While many methods exist to automatically correct anomalies and impute missing entries, challenges still exist on data with large spatial-temporal scales and shifting patterns. To address these challenges, we propose an online robust tensor recovery (OLRTR) method to preprocess streaming high-dimensional urban environmental datasets. A small-sized dictionary that captures the underlying patterns of the data is computed and constantly updated with new data. OLRTR enables online recovery for large-scale sensor networks that provide continuous data streams, with a lower computational memory usage compared to offline batch counterparts. In addition, we formulate the objective function so that OLRTR can detect structured outliers, such as faulty readings over a long period of time. We validate OLRTR on a synthetically degraded National Oceanic and Atmospheric Administration temperature dataset, and apply it to the Array of Things city-scale sensor network in Chicago, IL, showing superior results compared with several established online and batch-based low-rank decomposition methods.

Robust Tensor Recovery in Impulsive Noise Based on Correntropy and Hybrid Tensor Sparsity

Robust tensor recovery aims to reconstruct a multidimensional tensor from its observations contaminated by noise. In this brief, a new formulation based on correntropy and hybrid tensor sparsity measure is proposed for robust tensor recovery in the environment of impulsive noise. The robust correntropy measure has shown satisfactory robustness against large outliers in various scenarios recently. Meanwhile, in this formulation, the hybrid tensor sparsity measure combining the advantages of Tucker and CP tensor rank can better characterize the tensor sparsity. To solve the proposed formulation effectively, an efficient large-scale optimization algorithm is derived based on the framework of alternating direction method of multipliers (ADMM) and half-quadratic optimization technique. The results of multispectral image (MSI) data recovery indicate that the proposed algorithm can achieve robust tensor recovery in the environment of impulsive noise.

Nonlocal robust tensor recovery with nonconvex regularization * *The research of Bai was supported in part by the National Natural Science Foundation of China under Grant No. 11971159. The research of Ng was supported in part by the HKRGC GRF 12306616, 12200317, 12300218, 12300519 and 17201020, and HKU Grant No. 104005583. The research of Zhang was supported in part by

The robust tensor recovery problem consists in reconstructing a tensor from a sample of entries corrupted by noise, which has attracted great interest in a wide range of practical situations such as image processing and computer vision. In this paper, we study robust tensor recovery for third-order tensors with different degradations, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously. In contrast to traditional approaches based on the tensor nuclear norm penalty for the low-rank component and the tensor ℓ 1 norm penalty for the sparse component, we propose a nonlocal robust low-rank tensor recovery model with nonconvex regularization (NRTRM) to explore the global low-rankness and nonlocal self-similarity of the underlying tensor. The NRTRM method is first to extract similar patched-tubes to form a third-order sub-tensor. Then a class of nonconvex low-rank penalties and nonconvex sparse penalties are employed to explore the low-rank component and the sparse corruptions for such sub-tensor, respectively. Moreover, a proximal alternating linearized minimization algorithm is developed to solve the resulting model in each group and its convergence is established under very mild conditions. Extensive numerical experiments on both multispectral images and video datasets demonstrate the superior performance of NRTRM in comparison with several state-of-the-art methods.

Robust Tensor Recovery with Fiber Outliers for Traffic Events

Event detection is gaining increasing attention in smart cities research. Large-scale mobility data serves as an important tool to uncover the dynamics of urban transportation systems, and more often than not the dataset is incomplete. In this article, we develop a method to detect extreme events in large traffic datasets, and to impute missing data during regular conditions. Specifically, we propose a robust tensor recovery problem to recover low-rank tensors under fiber-sparse corruptions with partial observations, and use it to identify events, and impute missing data under typical conditions. Our approach is scalable to large urban areas, taking full advantage of the spatio-temporal correlations in traffic patterns. We develop an efficient algorithm to solve the tensor recovery problem based on the alternating direction method of multipliers (ADMM) framework. Compared with existing l 1 norm regularized tensor decomposition methods, our algorithm can exactly recover the values of uncorrupted fibers of a low-rank tensor and find the positions of corrupted fibers under mild conditions. Numerical experiments illustrate that our algorithm can achieve exact recovery and outlier detection even with missing data rates as high as 40% under 5% gross corruption, depending on the tensor size and the Tucker rank of the low rank tensor. Finally, we apply our method on a real traffic dataset corresponding to downtown Nashville, TN and successfully detect the events like severe car crashes, construction lane closures, and other large events that cause significant traffic disruptions.

Robust Long-Term Spectrum Prediction With Missing Values and Sparse Anomalies

Due to an increasingly instrumental role in dynamic spectrum access, spectrum prediction causes extensive concern. Recently, the long-term spectrum prediction scheme based on tensor completion (LSP-TC) was proposed, which performs prediction over a time–frequency-day spectrum tensor model. Nevertheless, LSP-TC suffers from missing data and sparse anomalies. In this paper, we show that the low-rank property of the tensor model may be destroyed by sparse anomalies, resulting in an invalid tensor completion. Therefore, robust tensor recovery (RTR) is introduced to fill the missing data and separate anomalies from the observation data. On the other hand, the regularity of the data sequences may also be severely affected by anomalies and missing data. To this end, a new prediction scheme named robust long-term spectrum prediction (RLSP) is proposed. Specifically, it alternatively performs prediction on partial data sequences (also referred to as pre-fill operation) and RTR in an iterative way, which not only reduces the damage to data sequence regularity brought by anomalies and missing data but also improves the accuracy of spectrum prediction. Finally, simulations based on the synthesis of data and real-world satellite spectrum data are given to show the superiority of our proposed RLSP over LSP-TC.

Fast Tensor Factorization for Accurate Internet Anomaly Detection

Detecting anomalous traffic is a critical task for advanced Internet management. Many anomaly detection algorithms have been proposed recently. However, constrained by their matrix-based traffic data model, existing algorithms often suffer from low accuracy in anomaly detection. To fully utilize the multi-dimensional information hidden in the traffic data, this paper takes the initiative to investigate the potential and methodologies of performing tensor factorization for more accurate Internet anomaly detection. More specifically, we model the traffic data as a three-way tensor and formulate the anomaly detection problem as a robust tensor recovery problem with the constraints on the rank of the tensor and the cardinality of the anomaly set. These constraints, however, make the problem extremely hard to solve. Rather than resorting to the convex relaxation at the cost of low detection performance, we propose TensorDet to solve the problem directly and efficiently. To improve the anomaly detection accuracy and tensor factorization speed, TensorDet exploits the factorization structure with two novel techniques, sequential tensor truncation and two-phase anomaly detection. We have conducted extensive experiments using Internet traffic trace data Abilene and GEANT. Compared with the state of art algorithms for tensor recovery and matrix-based anomaly detection, TensorDet can achieve significantly lower false positive rate and higher true positive rate. Particularly, benefiting from our well designed algorithm to reduce the computation cost of tensor factorization, the tensor factorization process in TensorDet is 5 (Abilene) and 13 (GEANT) times faster than that of the traditional Tucker decomposition solution.

Robust Low-Rank Tensor Recovery: Models and Algorithms

Robust tensor recovery plays an instrumental role in robustifying tensor decompositions for multilinear data analysis against outliers, gross corruptions and missing values and has a diverse array of applications. In this paper, we study the problem of robust low-rank tensor recovery in a convex optimization framework, drawing upon recent advances in robust Principal Component Analysis and tensor completion. We propose tailored optimization algorithms with global convergence guarantees for solving both the constrained and the Lagrangian formulations of the problem. These algorithms are based on the highly efficient alternating direction augmented Lagrangian and accelerated proximal gradient methods. We also propose a nonconvex model that can often improve the recovery results from the convex models. We investigate the empirical recoverability properties of the convex and nonconvex formulations and compare the computational performance of the algorithms on simulated data. We demonstrate through a number of real applications the practical effectiveness of this convex optimization framework for robust low-rank tensor recovery.