Huyan Huang

Huyan Huang, Yipeng Liu, Ce Zhu

Coupled tensor decomposition reveals the joint data structure by incorporating priori knowledge that come from the latent coupled factors. The tensor ring (TR) decomposition is invariant under the permutation of tensors with different mode properties, which ensures the uniformity of decomposed factors and mode attributes. The TR has powerful expression ability and achieves success in some multi-dimensional data processing applications. To let coupled tensors help each other for missing component estimation, in this paper we utilize TR for coupled completion by sharing parts of the latent factors. The optimization model for coupled TR completion is developed with a novel Frobenius norm. It is solved by the block coordinate descent algorithm which efficiently solves a series of quadratic problems resulted from sampling pattern. The excess risk bound for this optimization model shows the theoretical performance enhancement in comparison with other coupled nuclear norm based methods. The proposed method is validated on numerical experiments on synthetic data, and experimental results on real-world data demonstrate its superiority over the state-of-the-art methods in terms of recovery accuracy.

Huyan Huang, Yipeng Liu, Ce Zhu

Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical ones. Compared with TT and TR, the projected entangled pair state (PEPS), which is also called tensor grid (TG), allows more interactions between different dimensions, and may lead to more compact representation. In this paper, we propose to perform image completion based on low-rank tensor grid. A two-stage density matrix renormalization group algorithm is used for initialization of TG decomposition, which consists of multiple TT decompositions. The latent TG factors can be alternatively obtained by solving alternating least squares problems. To further improve the computational efficiency, a multi-linear matrix factorization for low rank TG completion is developed by using parallel matrix factorization. Experimental results on synthetic data and real-world images show the proposed methods outperform the existing ones in terms of recovery accuracy.

Huyan Huang, Yipeng Liu, Ce Zhu

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 dimensional size n and TR rank r can be exactly recovered with high probability by solving a convex optimization program, given n^{d/2} r^2 ln^7(n^{d/2})samples. 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.