Tensor train factorization under noisy and incomplete data with automatic rank estimation.

• Gaussian-product-Gamma prior with sparsity analysis for TT representation. • Variational Inference algorithm with automatic rank selection. • Complexity analysis and efficiency improvement for the algorithm. • Extensive experimental results on synthetic data and real-world datasets. As a powerful tool in analyzing multi-dimensional data, tensor train (TT) decomposition shows superior performance compared to other tensor decomposition formats. Existing TT decomposition methods, however, either easily overfit with noise, or require substantial fine-tuning to strike a balance between recovery accuracy and model complexity. To avoid the above shortcomings, this paper treats the TT decomposition in a fully Bayesian perspective, which includes automatic TT rank determination and noise power estimation. Theoretical justification on adopting the Gaussian-product-Gamma priors for inducing sparsity on the slices of the TT cores is provided, thus allowing the model complexity to be automatically determined even when the observed tensor data is noisy and contains many missing values. Furthermore, using the variational inference framework, an effective learning algorithm on the probabilistic model parameters is derived. Simulations on synthetic data demonstrate that the proposed algorithm accurately recovers the underlying TT structure from incomplete noisy observations. Further experiments on image and video data also show its superior performance to other existing TT decomposition algorithms. [ABSTRACT FROM AUTHOR]