Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trains
Department of Electrical and Electronic Engineering, The University of Hong Kong（香港大学电子工程系）；Delft Center for Systems and Control, Delft University of Technology（代尔夫特理工大学系统与控制中心）
Ching-Yun Ko, Kim Batselier, Wenjian Yu, Ngai Wong
备注：
13 pages. Source code and supplemental materials are available via: https://github.com/IRENEKO/TTC Updates 11/13: included more comparisons and experimental results
We propose a new tensor completion method based on tensor trains. The to-be-completed tensor is modeled as a low-rank tensor train, where we use the known tensor entries and their coordinates to update the tensor train. A novel tensor train initialization procedure is proposed specifically for image and video completion, which is demonstrated to ensure fast convergence of the completion algorithm. The tensor train framework is also shown to easily accommodate Total Variation and Tikhonov regularization due to their low-rank tensor train representations. Image and video inpainting experiments verify the superiority of the proposed scheme in terms of both speed and scalability, where a speedup of up to 155X is observed compared to state-of-the-art tensor completion methods at a similar accuracy. Moreover, we demonstrate the proposed scheme is especially advantageous over existing algorithms when only tiny portions (say, 1%) of the to-be-completed images/videos are known.
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…
Alternating minimization algorithms for graph regularized tensor completion
Huawei European Research Institute（华为欧洲研究院）；LISN, INRIA, Université Paris-Saclay（法国国家信息与自动化研究所，巴黎萨克雷大学）；LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院科学与工程计算国家重点实验室）；Department of INMA, Université catholique de Louvain（鲁汶天主教大学）
Xiamen University（厦门大学）；University of Science and Technology of China（中国科学技术大学）；Hong Kong University of Science and Technology（香港科技大学）；Xiamen University of Technology（厦门理工学院）
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 …
Completion of High Order Tensor Data with Missing Entries via Tensor-train Decomposition
Saitama Institute of Technology（埼玉工业大学）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心（AIP））；Guangdong University of Technology（广东工业大学）
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.
Standard regularization methods that are used to compute solutions to ill-posed inverse problems require knowledge of the forward model. In many real-life applications, the forward model is not known, but training data is readily available. In this paper, we develop a new framework that uses training data, as a substitute for knowledge of the forward model, to compute an optimal low-rank regularized inverse matrix directly, allowing for very fast computation of a regularized solution. We consider a statistical framework based on Bayes and empirical Bayes risk minimization to analyze theoretica…
High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format
Graduate School of Engineering, Saitama Institute of Technology, Japan（日本埼玉科技大学工学研究科）；Tensor Learning Unit, RIKEN Center for Advanced Intelligence Project (AIP), Japan（日本RIKEN先进人工智能项目（AIP）张量学习单元）；School of Automation, Guangdong University of Technology, China（广东技术大学自动化学院）；School of Computer Science and Technology, Hangzhou Dianzi University, China（杭州电子科技大学计算机科学与技术学院）
We propose a new method for low-rank approximation of Moore-Penrose pseudoinverses (MPPs) of large-scale matrices using tensor networks. The computed pseudoinverses can be useful for solving or preconditioning of large-scale overdetermined or underdetermined systems of linear equations. The computation is performed efficiently and stably based on the modified alternating least squares (MALS) scheme using low-rank tensor train (TT) decompositions and tensor network contractions. The formulated large-scale optimization problem is reduced to sequential smaller-scale problems for which any standar…
High-order Tensor Completion for Data Recovery via Sparse Tensor-train Optimization
Graduate School of Engineering, Saitama Institute of Technology, Japan（埼玉工科大学工学研究科）；Tensor Learning Unit, RIKEN Center for Advanced Intelligence Project (AIP), Japan（日本先进情报项目（AIP）RIKEN智能学习单元）；School of Automation, Guangdong University of Technology, China（广东技术大学自动化学院）；School of Computer Science and Technology, Hangzhou Dianzi University, China（杭州电子科技大学计算机科学与技术学院）
