Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known. &nbsp Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.
The work @cite_26 generalizes to the tensor case, and to recover the (approximately) low-rank tensor @math , it proposes to solve where @math are preselected weights satisfying @math . Different from our model , the problem is convex, and in @cite_26 , various methods are applied to solve it such as block coordinate descent method, proximal gradient method, and alternating direction method of multiplier (ADMM). The model utilizes low-rankness of all mode unfoldings of the tensor, and as demonstrated in @cite_26 , it can significantly improve the solution quality over that obtained by solving , where the matrix @math corresponds to some mode unfolding of the tensor.
@cite_26
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In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependant relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between FaLRTC and HaLRTC the former is more efficient to obtain a low accuracy solution and the latter is preferred if a high-accuracy solution is desired.