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We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis, which aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called robust tensor CUR decompositions (RTCUR), for large-scale nonconvex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets. 

Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations. 

Stress is a common problem in modern life that can bring both psychological and physical disorder. Wearable sensors are commonly used to study the relationship between physical records and mental status. Although sensor data generated by wearable devices provides an opportunity to identify stress in people for predictive medicine, in practice, the data are typically complicated and vague and also often fragmented. In this paper, we propose DataCompletion with Diurnal Regularizers (DCDR) and TemporallyHierarchical Attention Network (THAN) to address the fragmented data issue and predict human stress level with recovered sensor data. We model fragmentation as a sparsity issue. The nuclear norm minimization method based on the low-rank assumption is first applied to derive unobserved sensor data with diurnal patterns of human behaviors. A hierarchical recurrent neural network with the attention mechanism then models temporally structural information in the reconstructed sensor data, thereby inferring the predicted stress level. Data for this study were from 75 undergraduate students (taken from a sample of a larger study) who provided sensor data from smart wristbands. They also completed weekly stress surveys as ground-truth labels about their stress levels. This survey lasted 12 weeks and the sensor records are also in this period. The experimental results demonstrate that our approach significantly outperforms conventional methods in both data completion and stress level prediction. Moreover, an in-depth analysis further shows the effectiveness and robustness of our approach.