Coupled Tensor Completion via Low-rank Tensor Ring

The coupled tensor decomposition aims to reveal the latent data structure which may share common factors. Using the recently proposed tensor ring decomposition, in this paper we propose a non-convex method by alternately optimizing the latent factors. We provide an excess risk bound for the proposed alternating minimization model, which shows the improvement in completion performance. The proposed algorithm is validated on synthetic data.

Robust Tensor Recovery using Low-Rank Tensor Ring

Robust tensor completion recoveries the low-rank and sparse parts from its partially observed entries. In this paper, we propose the robust tensor ring completion (RTRC) model and rigorously analyze its exact recovery guarantee via TR-unfolding scheme, and the result is consistent with that of matrix case. We propose the algorithms for tensor ring robust principle component analysis (TRRPCA) and RTCR using the alternating direction method of multipliers (ADMM). The numerical experiment demonstrates that the proposed method outperforms the state-of-the-art ones in terms of recovery accuracy.

Provable Model for Tensor Ring Completion

Tensor completion aims to recover a multi-dimensional array from its incomplete observations. The recently proposed tensor ring (TR) decomposition has powerful representation ability and shows promising performance in tensor completion, though they suffer from lack of theoretical guarantee. In this paper, we rigorously analyze the sample complexity of TR completion and find it also possesses the balance characteristic, which is consistent with the result of matrix completion. Inspired by this property we propose a nuclear norm minimization model and solve it by the alternating direction method of multipliers (ADMM). The experiments on synthetic data verify the theoretic analysis, and the numerical results of real-world data demonstrate that the proposed method gains great performance improvement in tensor completion compared with the state-of-the-art ones.

Tensor Grid Decomposition with Application to Tensor Completion

The recently prevalent tensor train (TT) and tensor ring (TR) decompositions can be graphically interpreted as (locally) linear interconnected latent factors and possess exponential decay of correlation. The projected entangled pair state (PEPS, also called two-dimensional TT) extends the spatial dimension of TT and its polycyclic structure can be considered as a square grid. Compared with TT, its algebraic decay of correlation means the enhancement of interaction between tensor modes. In this paper we adopt the PEPS and develop a tensor grid (TG) decomposition with its efficient realization termed splitting singular value decomposition (SSVD). By utilizing the alternating least squares (ALS) a method called TG-ALS is used to interpolate the missing entries of a tensor from its partial observations. Different kinds of data are used in the experiments, including synthetic data, color images and real-world videos. Experimental results demonstrate that the TG has much power of representation than TT and TR.