Multidimensional Data Processing With Bayesian Inference via Structural Block Decomposition.

How to handle large multidimensional datasets, such as hyperspectral images and video information, efficiently and effectively plays a critical role in big-data processing. The characteristics of low-rank tensor decomposition in recent years demonstrate the essentials in describing the tensor rank, which often leads to promising approaches. However, most current tensor decomposition models consider the rank-1 component simply to be the vector outer product, which may not fully capture the correlated spatial information effectively for large-scale and high-order multidimensional datasets. In this article, we develop a new novel tensor decomposition model by extending it to the matrix outer product or called Bhattacharya-Mesner product, to form an effective dataset decomposition. The fundamental idea is to decompose tensors structurally in a compact manner as much as possible while retaining data spatial characteristics in a tractable way. By incorporating the framework of the Bayesian inference, a new tensor decomposition model on the subtle matrix unfolding outer product is established for both tensor completion and robust principal component analysis problems, including hyperspectral image completion and denoising, traffic data imputation, and video background subtraction. Numerical experiments on real-world datasets demonstrate the highly desirable effectiveness of the proposed approach.

Bayesian Dictionary Learning on Robust Tubal Transformed Tensor Factorization.

The recent study on tensor singular value decomposition (t-SVD) that performs the Fourier transform on the tubes of a third-order tensor has gained promising performance on multidimensional data recovery problems. However, such a fixed transformation, e.g., discrete Fourier transform and discrete cosine transform, lacks being self-adapted to the change of different datasets, and thus, it is not flexible enough to exploit the low-rank and sparse property of the variety of multidimensional datasets. In this article, we consider a tube as an atom of a third-order tensor and construct a data-driven learning dictionary from the observed noisy data along the tubes of the given tensor. Then, a Bayesian dictionary learning (DL) model with tensor tubal transformed factorization, aiming to identify the underlying low-tubal-rank structure of the tensor effectively via the data-adaptive dictionary, is developed to solve the tensor robust principal component analysis (TRPCA) problem. With the defined pagewise tensor operators, a variational Bayesian DL algorithm is established and updates the posterior distributions instantaneously along the third dimension to solve the TPRCA. Extensive experiments on real-world applications, such as color image and hyperspectral image denoising and background/foreground separation problems, demonstrate both effectiveness and efficiency of the proposed approach in terms of various standard metrics.