Deep Plug-and-play Prior for Low-rank Tensor Completion
University of Electronic Science and Technology of China（电子科技大学）；Southwestern University of Finance and Economics（西南财经大学）；Xi’an Jiaotong University（西安交通大学）
Xi-Le Zhao, Wen-Hao Xu, Tai-Xiang Jiang, Yao Wang, Michael Ng
Multi-dimensional images, such as color images and multi-spectral images, are highly correlated and contain abundant spatial and spectral information. However, real-world multi-dimensional images are usually corrupted by missing entries. By integrating deterministic low-rankness prior to the data-driven deep prior, we suggest a novel regularized tensor completion model for multi-dimensional image completion. In the objective function, we adopt the newly emerged tensor nuclear norm (TNN) to characterize the global low-rankness prior of the multi-dimensional images. We also formulate an implicit regularizer by plugging into a denoising neural network (termed as deep denoiser), which is convinced to express the deep image prior learned from a large number of natural images. The resulting model can be solved by the alternating directional method of multipliers algorithm under the plug-and-play (PnP) framework. Experimental results on color images, videos, and multi-spectral images demonstrate that the proposed method can recover both the global structure and fine details very well and achieve superior performance over competing methods in terms of quality metrics and visual effects.
Plug-and-Play optimization recently emerged as a powerful technique for solving inverse problems by plugging a denoiser into a classical optimization algorithm. The denoiser accounts for the regularization and therefore implicitly determines the prior knowledge on the data, hence replacing typical handcrafted priors. In this paper, we extend the concept of plug-and-play optimization to use denoisers that can be parameterized for non-constant noise variance. In that aim, we introduce a preconditioning of the ADMM algorithm, which mathematically justifies the use of such an adjustable denoiser. …
Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor data which only approximately fulfils the low-rank requirement. To address these two issues, we develop a data-adaptive tensor completion model which explicitly represents both the low-rank and non-low-rank structures in a latent tensor. Representing the non-low-rank structure separately from the low-rank one allows priors which capture the important distinc…
Existing low-rank tensor completion (LRTC) approaches aim at restoring a partially observed tensor by imposing a global low-rank constraint on the underlying completed tensor. However, such a global rank assumption suffers the trade-off between restoring the originally details-lacking parts and neglecting the potentially complex objects, making the completion performance unsatisfactory on both sides. To address this problem, we propose a novel and practical strategy for image restoration that restores the partially observed tensor in a coarse-to-fine (C2F) manner, which gets rid of such trade-…
This paper focus on recovering multi-dimensional data called tensor from randomly corrupted incomplete observation. Inspired by reweighted $l_1$ norm minimization for sparsity enhancement, this paper proposes a reweighted singular value enhancement scheme to improve tensor low tubular rank in the tensor completion process. An efficient iterative decomposition scheme based on t-SVD is proposed which improves low-rank signal recovery significantly. The effectiveness of the proposed method is established by applying to video completion problem, and experimental results reveal that the algorithm o…
In this paper, we propose a novel model to recover a low-rank tensor by simultaneously performing double nuclear norm regularized low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An block successive upper-bound minimization algorithm is applied to solve the model. Subsequence convergence of our algorithm can be established, and our algorithm converges to the coordinate-wise minimizers in some mild conditions. Several experiments on three types of public data sets show that our algorithm can recover a variety of low-rank tensors from significantly fewer sa…
University of Electronic Science and Technology of China（电子科技大学）；Xi’an Jiaotong University（西安交通大学）
In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on the convex relaxation by penalizing the weighted sum of nuclear norm of TR unfolding matrices. However, this method treats each singular value equally and neglects their physical meanings, which usually leads to suboptimal solutions in practice. In this paper, we propose to use the logdet-based function as a nonconvex smooth relaxation of the TR rank for tenso…
Low-rank tensor completion has been widely used in computer vision and machine learning. This paper develops a novel multi-modal core tensor factorization (MCTF) method combined with a tensor low-rankness measure and a better nonconvex relaxation form of this measure (NC-MCTF). 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 NC-…
Columbia University（哥伦比亚大学）；Purdue University（普渡大学）
In this letter, we study the deterministic sampling patterns for the completion of low rank matrix, when corrupted with a sparse noise, also known as robust matrix completion. We extend the recent results on the deterministic sampling patterns in the absence of noise based on the geometric analysis on the Grassmannian manifold. A special case where each column has a certain number of noisy entries is considered, where our probabilistic analysis performs very efficiently. Furthermore, assuming that the rank of the original matrix is not given, we provide an analysis to determine if the rank of …
University of Electronic Science and Technology of China（电子科技大学）；The University of Hong Kong（香港大学）
The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($\textrm{order} >3$). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with \emph{ket augmentation}, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completi…
As a real scenes sensing approach, depth information obtains the widespread applications. However, resulting from the restriction of depth sensing technology, the depth map captured in practice usually suffers terrible noise and missing values at plenty of pixels. In this paper, we propose a fast low-rank matrix completion via dual-graph regularization for depth restoration. Specifically, the depth restoration can be transformed into a low-rank matrix completion problem. In order to complete the low-rank matrix and restore it to the depth map, the proposed dual-graph method containing the loca…
College of Information Science and Electronic Engineering, Zhejiang University（浙江大学信息与电子工程学院）
Recently, low-rank tensor completion has become increasingly attractive in recovering incomplete visual data. Considering a color image or video as a three-dimensional (3D) tensor, existing studies have put forward several definitions of tensor nuclear norm. However, they are limited and may not accurately approximate the real rank of a tensor, and they do not explicitly use the low-rank property in optimization. It is proved that the recently proposed truncated nuclear norm (TNN) can replace the traditional nuclear norm, as an improved approximation to the rank of a matrix. In this paper, we …
Tensor ring (TR) decomposition has been successfully used to obtain the state-of-the-art performance in the visual data completion problem. However, the existing TR-based completion methods are severely non-convex and computationally demanding. In addition, the determination of the optimal TR rank is a tough work in practice. To overcome these drawbacks, we first introduce a class of new tensor nuclear norms by using tensor circular unfolding. Then we theoretically establish connection between the rank of the circularly-unfolded matrices and the TR ranks. We also develop an efficient tensor co…
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 smoo…
College of Information Science and Electronic Engineering, Zhejiang University（浙江大学信息科学与电子工程学院）；Zhejiang University（浙江大学）；College of Information Science and Electronic Engineering（信息科学与电子工程学院）；Hangzhou, China（中国杭州）
Currently, low-rank tensor completion has gained cumulative attention in recovering incomplete visual data whose partial elements are missing. By taking a color image or video as a three-dimensional (3D) tensor, previous studies have suggested several definitions of tensor nuclear norm. However, they have limitations and may not properly approximate the real rank of a tensor. Besides, they do not explicitly use the low-rank property in optimization. It is proved that the recently proposed truncated nuclear norm (TNN) can replace the traditional nuclear norm, as a better estimation to the rank …
Northwest A&F University（西北农林科技大学）；University of Macau（澳门大学）
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, w…
Hangzhou Dianzi University（杭州电子科技大学）；Ningbo University（宁波大学）；Huawei Technologies Co., Ltd.（华为技术有限公司）；The Hong Kong Polytechnic University（香港理工大学）
Recovering color images and videos from highly undersampled data is a fundamental and challenging task in face recognition and computer vision. By the multi-dimensional nature of color images and videos, in this paper, we propose a novel tensor completion approach, which is able to efficiently explore the sparsity of tensor data under the discrete cosine transform (DCT). Specifically, we introduce two ``sparse + low-rank'' tensor completion models as well as two implementable algorithms for finding their solutions. The first one is a DCT-based sparse plus weighted nuclear norm induced low-rank…
A novel tag completion algorithm is proposed in this paper, which is designed with the following features: 1) Low-rank and error s-parsity: the incomplete initial tagging matrix D is decomposed into the complete tagging matrix A and a sparse error matrix E. However, instead of minimizing its nuclear norm, A is further factor-ized into a basis matrix U and a sparse coefficient matrix V, i.e. D=UV+E. This low-rank formulation encapsulating sparse coding enables our algorithm to recover latent structures from noisy initial data and avoid performing too much denoising; 2) Local reconstruction stru…
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）；Imperial College London（帝国理工学院）；Hangzhou Dianzi University（杭州电子科技大学）；Guangdong University of Technology（广东工业大学）
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather challenging to find the optimal TR decomposition. In this paper, by exploiting the low-rank structure of the TR latent space, we propose a novel tensor completion method which is robust to model selection. In contrast to imposing the low-rank constraint on the data space, we introd…
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）
