Tensor Matched Subspace Detection

The problem of testing whether a signal lies within a given subspace, also named matched subspace detection, has been well studied when the signal is represented as a vector. However, the matched subspace detection methods based on vectors can not be applied to the situations that signals are naturally represented as multi-dimensional data arrays or tensors. Considering that tensor subspaces and orthogonal projections onto these subspaces are well defined in the recently proposed transform-based tensor model, which motivates us to investigate the problem of matched subspace detection in high dimensional case. In this paper, we propose an approach for tensor matched subspace detection based on the transform-based tensor model with tubal-sampling and elementwise-sampling, respectively. First, we construct estimators based on tubal-sampling and elementwise-sampling to estimate the energy of a signal outside a given subspace of a third-order tensor and then give the probability bounds of our estimators, which show that our estimators work effectively when the sample size is greater than a constant. Secondly, the detectors both for noiseless data and noisy data are given, and the corresponding detection performance analyses are also provided. Finally, based on discrete Fourier transform (DFT) and discrete cosine transform (DCT), the performance of our estimators and detectors are evaluated by several simulations, and simulation results verify the effectiveness of our approach

Tensor Completion Algorithms in Big Data Analytics

Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer vision, signal processing, and neuroscience. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. We characterize these advances from four perspectives: general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume), and dynamic tensor completion algorithms (velocity). Further, we identify several tensor completion applications on real-world data-driven problems and present some common experimental frameworks popularized in the literature. Our goal is to summarize these popular methods and introduce them to researchers and practitioners for promoting future research and applications. We conclude with a discussion of key challenges and promising research directions in this community for future exploration

Non-recurrent Traffic Congestion Detection with a Coupled Scalable Bayesian Robust Tensor Factorization Model

Non-recurrent traffic congestion (NRTC) usually brings unexpected delays to commuters. Hence, it is critical to accurately detect and recognize the NRTC in a real-time manner. The advancement of road traffic detectors and loop detectors provides researchers with a large-scale multivariable temporal-spatial traffic data, which allows the deep research on NRTC to be conducted. However, it remains a challenging task to construct an analytical framework through which the natural spatial-temporal structural properties of multivariable traffic information can be effectively represented and exploited to better understand and detect NRTC. In this paper, we present a novel analytical training-free framework based on coupled scalable Bayesian robust tensor factorization (Coupled SBRTF). The framework can couple multivariable traffic data including traffic flow, road speed, and occupancy through sharing a similar or the same sparse structure. And, it naturally captures the high-dimensional spatial-temporal structural properties of traffic data by tensor factorization. With its entries revealing the distribution and magnitude of NRTC, the shared sparse structure of the framework compasses sufficiently abundant information about NRTC. While the low-rank part of the framework, expresses the distribution of general expected traffic condition as an auxiliary product. Experimental results on real-world traffic data show that the proposed method outperforms coupled Bayesian robust principal component analysis (coupled BRPCA), the rank sparsity tensor decomposition (RSTD), and standard normal deviates (SND) in detecting NRTC. The proposed method performs even better when only traffic data in weekdays are utilized, and hence can provide more precise estimation of NRTC for daily commuters