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Register a new userMulti-Tensor Network Representation for High-Order Tensor Completion
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Added by
Chang Nie
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This work studies the problem of high-dimensional data (referred to tensors) completion from partially observed samplings. We consider that a tensor is a superposition of multiple low-rank components. In particular, each component can be represented as multilinear connections over several latent factors and naturally mapped to a specific tensor network (TN) topology. In this paper, we propose a fundamental tensor decomposition (TD) framework: Multi-Tensor Network Representation (MTNR), which can be regarded as a linear combination of a range of TD models, e.g., CANDECOMP/PARAFAC (CP) decomposition, Tensor Train (TT), and Tensor Ring (TR). Specifically, MTNR represents a high-order tensor as the addition of multiple TN models, and the topology of each TN is automatically generated instead of manually pre-designed. For the optimization phase, an adaptive topology learning (ATL) algorithm is presented to obtain latent factors of each TN based on a rank incremental strategy and a projection error measurement strategy. In addition, we theoretically establish the fundamental multilinear operations for the tensors with TN representation, and reveal the structural transformation of MTNR to a single TN. Finally, MTNR is applied to a typical task, tensor completion, and two effective algorithms are proposed for the exact recovery of incomplete data based on the Alternating Least Squares (ALS) scheme and Alternating Direction Method of Multiplier (ADMM) framework. Extensive numerical experiments on synthetic data and real-world datasets demonstrate the effectiveness of MTNR compared with the start-of-the-art methods.
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Higher-order low-rank tensor arises in many data processing applications and has attracted great interests. Inspired by low-rank approximation theory, researchers have proposed a series of effective tensor completion methods. However, most of these methods directly consider the global low-rankness of underlying tensors, which is not sufficient for a low sampling rate; in addition, the single nuclear norm or its relaxation is usually adopted to approximate the rank function, which would lead to suboptimal solution deviated from the original one. To alleviate the above problems, in this paper, we propose a novel low-rank approximation of tensor multi-modes (LRATM), in which a double nonconvex $L_{gamma}$ norm is designed to represent the underlying joint-manifold drawn from the modal factorization factors of the underlying tensor. A block successive upper-bound minimization method-based algorithm is designed to efficiently solve the proposed model, and it can be demonstrated that our numerical scheme converges to the coordinatewise minimizers. Numerical results on three types of public multi-dimensional datasets have tested and shown that our algorithm can recover a variety of low-rank tensors with significantly fewer samples than the compared methods.
Tensor decomposition is a popular technique for tensor completion, However most of the existing methods are based on linear or shallow model, when the data tensor becomes large and the observation data is very small, it is prone to over fitting and the performance decreases significantly. To address this problem, the completion method for a tensor based on a Biased Deep Tensor Factorization Network (BDTFN) is proposed. This method can not only overcome the shortcomings of traditional tensor factorization, but also deal with complex non-linear data. Firstly, the horizontal and lateral tensors corresponding to the observed values of the input tensors are used as inputs and projected to obtain their horizontal (lateral) potential feature tensors. Secondly, the horizontal (lateral) potential feature tensors are respectively constructed into a multilayer perceptron network. Finally, the horizontal and lateral output tensors are fused by constructing a bilinear pooling layer. Tensor forward-propagation is composed of those three step, and its parameters are updated by tensor back-propagation using the multivariable chain rule. In this paper, we consider the large-scale 5-minute traffic speed data set and use it to address the missing data imputation problem for large-scale spatiotemporal traffic data. In addition, we compare the numerical performance of the proposed algorithm with those for state-of-the-art approaches on video recovery and color image recovery. Numerical experimental results illustrate that our approach is not only much more accurate than those state-of-the-art methods, but it also has high speed.
Many tensor-based data completion methods aim to solve image and video in-painting problems. But, all methods were only developed for a single dataset. In most of real applications, we can usually obtain more than one dataset to reflect one phenomenon, and all the datasets are mutually related in some sense. Thus one question raised whether such the relationship can improve the performance of data completion or not? In the paper, we proposed a novel and efficient method by exploiting the relationship among datasets for multi-video data completion. Numerical results show that the proposed method significantly improve the performance of video in-painting, particularly in the case of very high missing percentage.
In recent years, low-rank based tensor completion, which is a higher-order extension of matrix completion, has received considerable attention. However, the low-rank assumption is not sufficient for the recovery of visual data, such as color and 3D images, where the ratio of missing data is extremely high. In this paper, we consider smoothness constraints as well as low-rank approximations, and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data. The proposed method admits significant advantages, owing to the integration of smooth PARAFAC decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank. Thus, our proposed method is termed as smooth PARAFAC tensor completion (SPC). In order to impose the smoothness constraints, we employ two strategies, total variation (SPC-TV) and quadratic variation (SPC-QV), and invoke the corresponding algorithms for model learning. Extensive experimental evaluations on both synthetic and real-world visual data illustrate the significant improvements of our method, in terms of both prediction performance and efficiency, compared with many state-of-the-art tensor completion methods.
Low-rank tensor completion has been widely used in computer vision and machine learning. This paper develops a kind of multi-modal core tensor factorization (MCTF) method together with a tensor low-rankness measure and a better nonconvex relaxation form of it (NonMCTF). The proposed models encode low-rank insights for general tensors provided by Tucker and T-SVD, and thus are expected to simultaneously model spectral low-rankness in multiple orientations and accurately restore the data of intrinsic low-rank structure based on few observed entries. Furthermore, we study the MCTF and NonMCTF regularization minimization problem, and design an effective BSUM algorithm to solve them. This efficient solver can extend MCTF to various tasks, such as tensor completion. A series of experiments, including hyperspectral image (HSI), video and MRI completion, confirm the superior performance of the proposed method.
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