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Tensor completion recovers missing components of multi-way data. The existing methods use either the Tucker rank or the CANDECOMP/PARAFAC (CP) rank in low-rank tensor optimization for data completion. In fact, these two kinds of tensor ranks represent different high-dimensional data structures. In this paper, we propose to exploit the two kinds of data structures simultaneously for image recovery through jointly minimizing the CP rank and Tucker rank in the low-rank tensor approximation. We use the alternating direction method of multipliers (ADMM) to reformulate the optimization model with two tensor ranks into its two sub-problems, and each has only one tensor rank optimization. For the two main sub-problems in the ADMM, we apply rank-one tensor updating and weighted sum of matrix nuclear norms minimization methods to solve them, respectively. The numerical experiments on some image and video completion applications demonstrate that the proposed method is superior to the state-of-the-art methods.

Low-rank tensor completion recovers missing entries based on different tensor decompositions. Due to its outstanding performance in exploiting some higher-order data structure, low rank tensor ring has been applied in tensor completion. To further deal with its sensitivity to sparse component as it does in tensor principle component analysis, we propose robust tensor ring completion (RTRC), which separates latent low-rank tensor component from sparse component with limited number of measurements. The low rank tensor component is constrained by the weighted sum of nuclear norms of its balanced unfoldings, while the sparse component is regularized by its $\ell _1$ norm. We analyze the RTRC model and gives the exact recovery guarantee. The alternating direction method of multipliers is used to divide the problem into several sub-problems with fast solutions. In numerical experiments, we verify the recovery condition of the proposed method on synthetic data, and show the proposed method outperforms the state-of-the-art ones in terms of both accuracy and computational complexity in a number of real-world data based tasks, i.e., light-field image recovery, shadow removal in face images, and background extraction in color video.

Tensor completion recovers a multi-dimensional array from a limited number of measurements. Using the recently proposed tensor ring (TR) decomposition, in this paper we show that a d-order tensor of size n × ⋅⋅⋅ × n and TR rank [r, ... , r] can be exactly recovered with high probability by solving a convex optimization program, given O(n⌈d/2⌉r2ln 7(n⌈d/2⌉)) samples. In the optimization model, a weighted sum of nuclear norms of factors surrogates the TR rank. The proposed TR incoherence condition under which the result holds is similar to the matrix incoherence condition. The experiments on synthetic data verify the recovery guarantee for TR completion. Moreover, the experiments on real-world data show that our method improves the recovery performance compared with the state-of-the-art methods.

The emergence of large-scale data demands new regression models with multi-dimensional coefficient arrays, known as tensor regression models. The recently proposed tensor ring decomposition has interesting properties of enhanced representation and compression capability, cyclic permutation invariance and balanced tensor ring rank, which may lead to efficient computation and fewer parameters in regression problems. In this paper, a generally multi-linear tensor-on-tensor regression model is proposed that the coefficient array has a low-rank tensor ring structure, which is termed tensor ring ridge regression (TRRR). Two optimization models are developed for the TRRR problem and solved by different algorithms: the tensor factorization based one is solved by alternating least squares algorithm, and accelerated by a fast network contraction, while the rank minimization based one is addressed by the alternating direction method of multipliers algorithm. Comparative experiments, including Spatio-temporal forecasting tasks and 3D reconstruction of human motion capture data from its temporally synchronized video sequences, demonstrate the enhanced performance of our algorithms over existing state-of-the-art ones, especially in terms of training time.