One of the popular approaches for low-rank tensor completion is to use the latent trace norm as a low-rank regularizer. However, most of the existing works learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm which helps to learn a non-sparse combination of tensors. We develop a dual framework for solving the problem of latent trace norm regularized low-rank tensor completion. In this framework, we first show a novel characterization of the solution space with a novel factorization, and then, propose two scalable optimization formulations. The problems are shown to lie on a Cartesian product of Riemannian spectrahedron manifolds. We exploit the versatile Riemannian optimization framework for proposing computationally efficient trust-region algorithms. The experiments show the good performance of the proposed algorithms on several real-world data sets in different applications.
One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.
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.
We study the problem of learning a tensor from a set of linear measurements. A prominent methodology for this problem is based on a generalization of trace norm regularization, which has been used extensively for learning low rank matrices, to the tensor setting. In this paper, we highlight some limitations of this approach and propose an alternative convex relaxation on the Euclidean ball. We then describe a technique to solve the associated regularization problem, which builds upon the alternating direction method of multipliers. Experiments on one synthetic dataset and two real datasets indicate that the proposed method improves significantly over tensor trace norm regularization in terms of estimation error, while remaining computationally tractable.
Video completion is a computer vision technique to recover the missing values in video sequences by filling the unknown regions with the known information. In recent research, tensor completion, a generalization of matrix completion for higher order data, emerges as a new solution to estimate the missing information in video with the assumption that the video frames are homogenous and correlated. However, each video clip often stores the heterogeneous episodes and the correlations among all video frames are not high. Thus, the regular tenor completion methods are not suitable to recover the video missing values in practical applications. To solve this problem, we propose a novel spatially-temporally consistent tensor completion method for recovering the video missing data. Instead of minimizing the average of the trace norms of all matrices unfolded along each mode of a tensor data, we introduce a new smoothness regularization along video time direction to utilize the temporal information between consecutive video frames. Meanwhile, we also minimize the trace norm of each individual video frame to employ the spatial correlations among pixels. Different to previous tensor completion approaches, our new method can keep the spatio-temporal consistency in video and do not assume the global correlation in video frames. Thus, the proposed method can be applied to the general and practical video completion applications. Our method shows promising results in all evaluations on both 3D biomedical image sequence and video benchmark data sets.