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.