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.
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.
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.
Coupled norms have emerged as a convex method to solve coupled tensor completion. A limitation with coupled norms is that they only induce low-rankness using the multilinear rank of coupled tensors. In this paper, we introduce a new set of coupled norms known as coupled nuclear norms by constraining the CP rank of coupled tensors. We propose new coupled completion models using the coupled nuclear norms as regularizers, which can be optimized using computationally efficient optimization methods. We derive excess risk bounds for proposed coupled completion models and show that proposed norms lead to better performance. Through simulation and real-data experiments, we demonstrate that proposed norms achieve better performance for coupled completion compared to existing coupled norms.
We propose a set of convex low rank inducing norms for a coupled matrices and tensors (hereafter coupled tensors), which shares information between matrices and tensors through common modes. More specifically, we propose a mixture of the overlapped trace norm and the latent norms with the matrix trace norm, and then, we propose a new completion algorithm based on the proposed norms. A key advantage of the proposed norms is that it is convex and can find a globally optimal solution, while existing methods for coupled learning are non-convex. Furthermore, we analyze the excess risk bounds of the completion model regularized by our proposed norms which show that our proposed norms can exploit the low rankness of coupled tensors leading to better bounds compared to uncoupled norms. Through synthetic and real-world data experiments, we show that the proposed completion algorithm compares favorably with existing completion algorithms.