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### Statistical Performance of Convex Tensor Decomposition

We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as non-convex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex low-rank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice.

### Tensor Completion Algorithms in Big Data Analytics

Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in data mining, computer vision, signal processing, and neuroscience, etc. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. Towards a better comprehension and comparison of vast existing advances, we summarize and categorize them into four groups including general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume) and dynamic tensor completion algorithms (velocity). Besides, we introduce their applications on real-world data-driven problems and present an open-source package covering several widely used tensor decomposition and completion algorithms. Our goal is to summarize these popular methods and introduce them to researchers for promoting the research process in this field and give an available repository for practitioners. In the end, we also discuss some challenges and promising research directions in this community for future explorations.

### 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.

### On Polynomial Time Methods for Exact Low Rank Tensor Completion

In this paper, we investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. We show that a gradient descent algorithm with initial value obtained from a spectral method can, in particular, reconstruct a ${d\times d\times d}$ tensor of multilinear ranks $(r,r,r)$ with high probability from as few as $O(r^{7/2}d^{3/2}\log^{7/2}d+r^7d\log^6d)$ entries. In the case when the ranks $r=O(1)$, our sample size requirement matches those for nuclear norm minimization (Yuan and Zhang, 2016a), or alternating least squares assuming orthogonal decomposability (Jain and Oh, 2014). Unlike these earlier approaches, however, our method is efficient to compute, easy to implement, and does not impose extra structures on the tensor. Numerical results are presented to further demonstrate the merits of the proposed approach.

### 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.