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Register a new userLow-Rank Autoregressive Tensor Completion for Spatiotemporal Traffic Data Imputation
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Spatiotemporal traffic time series (e.g., traffic volume/speed) collected from sensing systems are often incomplete with considerable corruption and large amounts of missing values, preventing users from harnessing the full power of the data. Missing data imputation has been a long-standing research topic and critical application for real-world intelligent transportation systems. A widely applied imputation method is low-rank matrix/tensor completion; however, the low-rank assumption only preserves the global structure while ignores the strong local consistency in spatiotemporal data. In this paper, we propose a low-rank autoregressive tensor completion (LATC) framework by introducing textit{temporal variation} as a new regularization term into the completion of a third-order (sensor $times$ time of day $times$ day) tensor. The third-order tensor structure allows us to better capture the global consistency of traffic data, such as the inherent seasonality and day-to-day similarity. To achieve local consistency, we design the temporal variation by imposing an AR($p$) model for each time series with coefficients as learnable parameters. Different from previous spatial and temporal regularization schemes, the minimization of temporal variation can better characterize temporal generative mechanisms beyond local smoothness, allowing us to deal with more challenging scenarios such blackout missing. To solve the optimization problem in LATC, we introduce an alternating minimization scheme that estimates the low-rank tensor and autoregressive coefficients iteratively. We conduct extensive numerical experiments on several real-world traffic data sets, and our results demonstrate the effectiveness of LATC in diverse missing scenarios.
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Missing value problem in spatiotemporal traffic data has long been a challenging topic, in particular for large-scale and high-dimensional data with complex missing mechanisms and diverse degrees of missingness. Recent studies based on tensor nuclear norm have demonstrated the superiority of tensor learning in imputation tasks by effectively characterizing the complex correlations/dependencies in spatiotemporal data. However, despite the promising results, these approaches do not scale well to large data tensors. In this paper, we focus on addressing the missing data imputation problem for large-scale spatiotemporal traffic data. To achieve both high accuracy and efficiency, we develop a scalable tensor learning model -- Low-Tubal-Rank Smoothing Tensor Completion (LSTC-Tubal) -- based on the existing framework of Low-Rank Tensor Completion, which is well-suited for spatiotemporal traffic data that is characterized by multidimensional structure of location$times$ time of day $times$ day. In particular, the proposed LSTC-Tubal model involves a scalable tensor nuclear norm minimization scheme by integrating linear unitary transformation. Therefore, tensor nuclear norm minimization can be solved by singular value thresholding on the transformed matrix of each day while the day-to-day correlation can be effectively preserved by the unitary transform matrix. We compare LSTC-Tubal with state-of-the-art baseline models, and find that LSTC-Tubal can achieve competitive accuracy with a significantly lower computational cost. In addition, the LSTC-Tubal will also benefit other tasks in modeling large-scale spatiotemporal traffic data, such as network-level traffic forecasting.
Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical ones. Compared with TT and TR, the projected entangled pair state (PEPS), which is also called tensor grid (TG), allows more interactions between different dimensions, and may lead to more compact representation. In this paper, we propose to perform image completion based on low-rank tensor grid. A two-stage density matrix renormalization group algorithm is used for initialization of TG decomposition, which consists of multiple TT decompositions. The latent TG factors can be alternatively obtained by solving alternating least squares problems. To further improve the computational efficiency, a multi-linear matrix factorization for low rank TG completion is developed by using parallel matrix factorization. Experimental results on synthetic data and real-world images show the proposed methods outperform the existing ones in terms of recovery accuracy.
Low-rank tensor completion recovers missing entries based on different tensor decompositions. Due to its outstanding performance in exploiting some higher-order data structure, low rank tensor ring has been applied in tensor completion. To further deal with its sensitivity to sparse component as it does in tensor principle component analysis, we propose robust tensor ring completion (RTRC), which separates latent low-rank tensor component from sparse component with limited number of measurements. The low rank tensor component is constrained by the weighted sum of nuclear norms of its balanced unfoldings, while the sparse component is regularized by its l1 norm. We analyze the RTRC model and gives the exact recovery guarantee. The alternating direction method of multipliers is used to divide the problem into several sub-problems with fast solutions. In numerical experiments, we verify the recovery condition of the proposed method on synthetic data, and show the proposed method outperforms the state-of-the-art ones in terms of both accuracy and computational complexity in a number of real-world data based tasks, i.e., light-field image recovery, shadow removal in face images, and background extraction in color video.
Tensor completion refers to the task of estimating the missing data from an incomplete measurement or observation, which is a core problem frequently arising from the areas of big data analysis, computer vision, and network engineering. Due to the multidimensional nature of high-order tensors, the matrix approaches, e.g., matrix factorization and direct matricization of tensors, are often not ideal for tensor completion and recovery. In this paper, we introduce a unified low-rank and sparse enhanced Tucker decomposition model for tensor completion. Our model possesses a sparse regularization term to promote a sparse core tensor of the Tucker decomposition, which is beneficial for tensor data compression. Moreover, we enforce low-rank regularization terms on factor matrices of the Tucker decomposition for inducing the low-rankness of the tensor with a cheap computational cost. Numerically, we propose a customized ADMM with enough easy subproblems to solve the underlying model. It is remarkable that our model is able to deal with different types of real-world data sets, since it exploits the potential periodicity and inherent correlation properties appeared in tensors. A series of computational experiments on real-world data sets, including internet traffic data sets, color images, and face recognition, demonstrate that our model performs better than many existing state-of-the-art matricization and tensorization approaches in terms of achieving higher recovery accuracy.
Missing value imputation is a fundamental problem in spatiotemporal modeling, from motion tracking to the dynamics of physical systems. Deep autoregressive models suffer from error propagation which becomes catastrophic for imputing long-range sequences. In this paper, we take a non-autoregressive approach and propose a novel deep generative model: Non-AutOregressive Multiresolution Imputation (NAOMI) to impute long-range sequences given arbitrary missing patterns. NAOMI exploits the multiresolution structure of spatiotemporal data and decodes recursively from coarse to fine-grained resolutions using a divide-and-conquer strategy. We further enhance our model with adversarial training. When evaluated extensively on benchmark datasets from systems of both deterministic and stochastic dynamics. NAOMI demonstrates significant improvement in imputation accuracy (reducing average prediction error by 60% compared to autoregressive counterparts) and generalization for long range sequences.
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