Online Robust Low-Rank Tensor Modeling for Streaming Data Analysis

Robust Online Tensor Completion for IoT Streaming Data Recovery

Reliable data measurement is considered to be one of the critical ingredients for variant Internet of Things (IoT) applications. Gaining full knowledge of measurement data is becoming increasingly crucial to ensure a satisfactory user experience. However, data missing and corruption are inevitable in practical applications, which motivates us to study how to accurately recover the missing IoT measurement data in the presence of outliers. The data recovery problem can be formulated as a tensor completion (TC) problem. Existing TC methods are built on the assumption that the rank of the underlying tensor is fixed, which is not suitable for long data sequences in practice. Consequently, based on the characteristics of IoT streaming data, we assume that the data tensor lies in time-varying subspace, and an accurate estimate of the rank is a prerequisite for filling the missing entries and achieving robustness of the variations in both rank and noise. We built up an updatable framework based on dynamic CANDECOMP/PARAFAC (CP) decomposition. In addition, an efficient algorithm, called temporal multi-aspect streaming (T-MUST), is introduced to solve the optimization problem that originates in our developed model. It is worth noting that the proposed algorithm allows time-varying tensor rank and enables the rank changes could be detected and tracked automatically. Theoretical analysis indicates that T-MUST enjoys a geometric convergence rate. Numerical experiments conducted on various synthetic and real-world datasets empirically validate the superiority of the proposed T-MUST in both efficiency and effectiveness.

Tracking online low-rank approximations of higher-order incomplete streaming tensors

In this paper, we propose two new provable algorithms for tracking online low-rank approximations of high-order streaming tensors with missing data. The first algorithm, dubbed adaptive Tucker decomposition (ATD), minimizes a weighted recursive least-squares cost function to obtain the tensor factors and the core tensor in an efficient way, thanks to an alternating minimization framework and a randomized sketching technique. Under the canonical polyadic (CP) model, the second algorithm, called ACP, is developed as a variant of ATD when the core tensor is imposed to be identity. Both algorithms are low-complexity tensor trackers that have fast convergence and low memory storage requirements. A unified convergence analysis is presented for ATD and ACP to justify their performance. Experiments indicate that the two proposed algorithms are capable of streaming tensor decomposition with competitive performance with respect to estimation accuracy and runtime on both synthetic and real data.

Online subspace learning and imputation by Tensor-Ring decomposition

This paper considers the completion problem of a partially observed high-order streaming data, which is cast as an online low-rank tensor completion problem. Though the online low-rank tensor completion problem has drawn lots of attention in recent years, most of them are designed based on the traditional decomposition method, such as CP and Tucker. Inspired by the advantages of Tensor Ring decomposition over the traditional decompositions in expressing high-order data and its superiority in missing values estimation, this paper proposes two online subspace learning and imputation methods based on Tensor Ring decomposition. Specifically, we first propose an online Tensor Ring subspace learning and imputation model by formulating an exponentially weighted least squares with Frobenium norm regularization of TR-cores. Then, two commonly used optimization algorithms, i.e. alternating recursive least squares and stochastic-gradient algorithms, are developed to solve the proposed model. Numerical experiments show that the proposed methods are more effective to exploit the time-varying subspace in comparison with the conventional Tensor Ring completion methods. Besides, the proposed methods are demonstrated to be superior to obtain better results than state-of-the-art online methods in streaming data completion under varying missing ratios and noise.

Low-Rank Tensor Train Coefficient Array Estimation for Tensor-on-Tensor Regression

The tensor-on-tensor regression can predict a tensor from a tensor, which generalizes most previous multilinear regression approaches, including methods to predict a scalar from a tensor, and a tensor from a scalar. However, the coefficient array could be much higher dimensional due to both high-order predictors and responses in this generalized way. Compared with the current low CANDECOMP/PARAFAC (CP) rank approximation-based method, the low tensor train (TT) approximation can further improve the stability and efficiency of the high or even ultrahigh-dimensional coefficient array estimation. In the proposed low TT rank coefficient array estimation for tensor-on-tensor regression, we adopt a TT rounding procedure to obtain adaptive ranks, instead of selecting ranks by experience. Besides, an l₂ constraint is imposed to avoid overfitting. The hierarchical alternating least square is used to solve the optimization problem. Numerical experiments on a synthetic data set and two real-life data sets demonstrate that the proposed method outperforms the state-of-the-art methods in terms of prediction accuracy with comparable computational complexity, and the proposed method is more computationally efficient when the data are high dimensional with small size in each mode.

Accurate vessel extraction via tensor completion of background layer in X-ray coronary angiograms

This paper proposes an effective method for accurately recovering vessel structures and intensity information from the X-ray coronary angiography (XCA) images of moving organs or tissues. Specifically, a global logarithm transformation of XCA images is implemented to fit the X-ray attenuation sum model of vessel/background layers into a low-rank, sparse decomposition model for vessel/background separation. The contrast-filled vessel structures are extracted by distinguishing the vessels from the low-rank backgrounds by using a robust principal component analysis and by constructing a vessel mask via Radon-like feature filtering plus spatially adaptive thresholding. Subsequently, the low-rankness and inter-frame spatio-temporal connectivity in the complex and noisy backgrounds are used to recover the vessel-masked background regions using tensor completion of all other background regions, while the twist tensor nuclear norm is minimized to complete the background layers. Finally, the method is able to accurately extract vessels' intensities from the noisy XCA data by subtracting the completed background layers from the overall XCA images. We evaluated the vessel visibility of resulting images on real X-ray angiography data and evaluated the accuracy of vessel intensity recovery on synthetic data. Experiment results show the superiority of the proposed method over the state-of-the-art methods.