This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global information of tensors thanks to its construction by a well-balanced matricization scheme. Two algorithms are proposed to solve the corresponding tensor completion problem. The first one called simple low-rank tensor completion via tensor train (SiLRTC-TT) is intimately related to minimizing the TT nuclear norm. The second one is based on a multilinear matrix factorization model to approximate the TT rank of the tensor and called tensor completion by parallel matrix factorization via tensor train (TMac-TT). These algorithms are applied to complete both synthetic and real world data tensors. Simulation results of synthetic data show that the proposed algorithms are efficient in estimating missing entries for tensors with either low Tucker rank or TT rank while Tucker-based algorithms are only comparable in the case of low Tucker rank tensors. When applied to recover color images represented by ninth-order tensors augmented from third-order ones, the proposed algorithms outperforms the Tucker-based algorithms.
We consider the low-rank tensor train completion problem when additional side information is available in the form of subspaces that contain the mode-$k$ fiber spans. We propose an algorithm based on Riemannian optimization to solve the problem. Numerical experiments show that the proposed algorithm requires far fewer known entries to recover the tensor compared to standard tensor train completion methods.
In this paper, we consider the rank-one tensor completion problem. We address the question of existence and uniqueness of the rank-one solution. In particular we show that the global uniqueness over the field of real numbers can be verified in a polynomial time. We give examples showing that there is an essential difference between the question of global uniqueness over the fields of real and complex numbers. Finally we briefly discuss the rank-one approximation problem for noisy observations.
In this paper, we describe a low-rank matrix completion method based on matrix decomposition. An incomplete matrix is decomposed into submatrices which are filled with a proposed trimming step and then are recombined to form a low-rank completed matrix. The divide-and-conquer approach can significantly reduce computation complexity and storage requirement. Moreover, the proposed decomposition method can be naturally incorporated into any existing matrix completion methods to attain further gain. Unlike most existing approaches, the proposed method is not based on norm minimization nor SVD deco…
This paper considers the completion problem for a tensor (also referred to as a multidimensional array) from limited sampling. Our greedy method is based on extending the low-rank approximation pursuit (LRAP) method for matrix completions to tensor completions. The method performs a tensor factorization using the tensor singular value decomposition (t-SVD) which extends the standard matrix SVD to tensors. The t-SVD leads to a notion of rank, called tubal-rank here. We want to recreate the data in tensors from low resolution samples as best we can here. To complete a low resolution tensor succe…
Low-rank quaternion tensor completion method, a novel approach to recovery color videos and images is proposed in this paper. We respectively reconstruct a color image and a color video as a quaternion matrix (second-order tensor) and a third-order quaternion tensor by encoding the red, green, and blue channel pixel values on the three imaginary parts of a quaternion. Different from some traditional models which treat color pixel as a scalar and represent color channels separately, whereas, during the quaternion-based reconstruction, it is significant that the inherent color structures of colo…
We consider the multi-objective optimization problem of choosing the bottom left block-entry of a block lower triangular matrix to minimize the ranks of all block sub-matrices. We provide a proof that there exists a simultaneous rank-minimizer by constructing the complete set of all minimizers.
Dimension reduction techniques are often used when the high-dimensional tensor has relatively low intrinsic rank compared to the ambient dimension of the tensor. The CANDECOMP/PARAFAC (CP) tensor completion is a widely used approach to find a low-rank approximation for a given tensor. In the tensor model, an $\ell_1$ regularized optimization problem was formulated with an appropriate choice of the regularization parameter. The choice of the regularization parameter is important in the approximation accuracy. However, the emergence of the large amount of data poses onerous computational burden …
The recent low-rank prior based models solve the tensor completion problem efficiently. However, these models fail to exploit the local patterns of tensors, which compromises the performance of tensor completion. In this paper, we propose a novel hierarchical prior regularized matrix factorization model for tensor completion. This model hierarchically incorporates the low-rank prior, total variation prior, and sparse coding prior into a matrix factorization, simultaneously characterizing both the global low-rank property and the local smoothness of tensors. For solving the proposed model, we u…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends on the values of the known entries. If the entries of the matrix are complex numbers, then for a fixed pattern of locations of specified and unspecified entries there is a unique completion rank which occurs with positive probability. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive prob…
University of Electronic Science and Technology of China（电子科技大学）
Tensor completion is a challenging problem with various applications. Many related models based on the low-rank prior of the tensor have been proposed. However, the low-rank prior may not be enough to recover the original tensor from the observed incomplete tensor. In this paper, we prose a tensor completion method by exploiting both the low-rank and sparse prior of tensor. Specifically, the tensor completion task can be formulated as a low-rank minimization problem with a sparse regularizer. The low-rank property is depicted by the tensor truncated nuclear norm based on tensor singular value …
We study alternating minimization for matrix completion in the simplest possible setting: completing a rank-one matrix from a revealed subset of the entries. We bound the asymptotic convergence rate by the variational characterization of eigenvalues of a reversible consensus problem. This leads to a polynomial upper bound on the asymptotic rate in terms of number of nodes as well as the largest degree of the graph of revealed entries.
In this paper we offer a review and bibliography of work on Hankel low-rank approximation and completion, with particular emphasis on how this methodology can be used for time series analysis and forecasting. We begin by describing possible formulations of the problem and offer commentary on related topics and challenges in obtaining globally optimal solutions. Key theorems are provided, and the paper closes with some expository examples.
As a new color image representation tool, quaternion has achieved excellent results in color image processing problems. In this paper, we propose a novel low-rank quaternion matrix completion algorithm to recover missing data of color image. Motivated by two kinds of low-rank approximation approaches (low-rank decomposition and nuclear norm minimization) in traditional matrix-based methods, we combine the two approaches in our quaternion matrix-based model. Furthermore, the nuclear norm of the quaternion matrix is replaced by the sum of Frobenius norm of its two low-rank factor quaternion matr…
A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated New…
Robust Principal Component Analysis (PCA) (Candes et al., 2011) and low-rank matrix completion (Recht et al., 2010) are extensions of PCA to allow for outliers and missing entries respectively. It is well-known that solving these problems requires a low coherence between the low-rank matrix and the canonical basis, since in the extreme cases -- when the low-rank matrix we wish to recover is also sparse -- there is an inherent ambiguity. However, the well-posedness issue in both problems is an even more fundamental one: in some cases, both Robust PCA and matrix completion can fail to have any s…
Matrix completion, the problem of completing missing entries in a data matrix with low dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog, that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted HOSVD algorithm for recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly w…
Non-convex approaches for low-rank tensor completion under tubal sampling
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Michigan State University（密歇根州立大学）；University of Central Florida（中佛罗里达大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
We consider the problem of low canonical polyadic (CP) rank tensor completion. A completion is a tensor whose entries agree with the observed entries and its rank matches the given CP rank. We analyze the manifold structure corresponding to the tensors with the given rank and define a set of polynomials based on the sampling pattern and CP decomposition. Then, we show that finite completability of the sampled tensor is equivalent to having a certain number of algebraically independent polynomials among the defined polynomials. Our proposed approach results in characterizing the maximum number …
