Dictionary Learning With Low-Rank Coding Coefficients for Tensor Completion

Online Sparse Representation Clustering for Evolving Data Streams

Data stream clustering can be performed to discover the patterns underlying continuously arriving sequences of data. A number of data stream clustering algorithms for finding clusters in arbitrary shapes and handling outliers, such as density-based clustering algorithms, have been proposed. However, these algorithms are often limited in their ability to construct and merge microclusters by measuring the Euclidean distances between high-dimensional data objects, e.g., transferring valuable knowledge from historical landmark windows to the current landmark window, and exploiting evolving subspace structures adaptively. We propose an online sparse representation clustering (OSRC) method to learn an affinity matrix for evaluating the relationships among high-dimensional data objects in evolving data streams. We first introduce a low-dimensional projection (LDP) into sparse representation to adaptively reduce the potential negative influence associated with the noise and redundancy contained in high-dimensional data. Then, we take advantage of the -norm optimization technique to choose the appropriate number of representative data objects and form a specific dictionary for sparse representation. The specific dictionary is integrated into sparse representation to adaptively exploit the evolving subspace structures of the high-dimensional data objects. Moreover, the data object representatives from the current landmark window can transfer valuable knowledge to the next landmark window. The experimental results based on a synthetic dataset and six benchmark datasets validate the effectiveness of the proposed method compared to that of state-of-the-art methods for data stream clustering.

Smooth Tensor Product for Tensor Completion

Low-rank tensor completion (LRTC) has shown promise in processing incomplete visual data, yet it often overlooks the inherent local smooth structures in images and videos. Recent advances in LRTC, integrating total variation regularization to capitalize on the local smoothness, have yielded notable improvements. Nonetheless, these methods are limited to exploiting local smoothness within the original data space, neglecting the latent factor space of tensors. More seriously, there is a lack of theoretical backing for the role of local smoothness in enhancing recovery performance. In response, this paper introduces an innovative tensor completion model that concurrently leverages the global low-rank structure of the original tensor and the local smooth structure of its factor tensors. Our objective is to learn a low-rank tensor that decomposes into two factor tensors, each exhibiting sufficient local smoothness. We propose an efficient alternating direction method of multipliers to optimize our model. Further, we establish generalization error bounds for smooth factor-based tensor completion methods across various decomposition frameworks. These bounds are significantly tighter than existing baselines. We conduct extensive inpainting experiments on color images, multispectral images, and videos, which demonstrate the efficacy and superiority of our method. Additionally, our approach shows a low sensitivity to hyper-parameter settings, enhancing its convenience and reliability for practical applications.

Multi-Dimensional Visual Data Restoration: Uncovering the Global Discrepancy in Transformed High-Order Tensor Singular Values

The recently proposed high-order tensor algebraic framework generalizes the tensor singular value decomposition (t-SVD) induced by the invertible linear transform from order-3 to order-d ( ). However, the derived order-d t-SVD rank essentially ignores the implicit global discrepancy in the quantity distribution of non-zero transformed high-order singular values across the higher modes of tensors. This oversight leads to suboptimal restoration in processing real-world multi-dimensional visual datasets. To address this challenge, in this study, we look in-depth at the intrinsic properties of practical visual data tensors, and put our efforts into faithfully measuring their high-order low-rank nature. Technically, we first present a novel order-d tensor rank definition. This rank function effectively captures the aforementioned discrepancy property observed in real visual data tensors and is thus called the discrepant t-SVD rank. Subsequently, we introduce a nonconvex regularizer to facilitate the construction of the corresponding discrepant t-SVD rank minimization regime. The results show that the investigated low-rank approximation has the closed-form solution and avoids dilemmas caused by the previous convex optimization approach. Based on this new regime, we meticulously develop two models for typical restoration tasks: high-order tensor completion and high-order tensor robust principal component analysis. Numerical examples on order-4 hyperspectral videos, order-4 color videos, and order-5 light field images substantiate that our methods outperform state-of-the-art tensor-represented competitors. Finally, taking a fundamental order-3 hyperspectral tensor restoration task as an example, we further demonstrate the effectiveness of our new rank minimization regime for more practical applications. The source codes of the proposed methods are available at https://github.com/CX-He/DTSVD.git.

Bayesian Dictionary Learning on Robust Tubal Transformed Tensor Factorization

The recent study on tensor singular value decomposition (t-SVD) that performs the Fourier transform on the tubes of a third-order tensor has gained promising performance on multidimensional data recovery problems. However, such a fixed transformation, e.g., discrete Fourier transform and discrete cosine transform, lacks being self-adapted to the change of different datasets, and thus, it is not flexible enough to exploit the low-rank and sparse property of the variety of multidimensional datasets. In this article, we consider a tube as an atom of a third-order tensor and construct a data-driven learning dictionary from the observed noisy data along the tubes of the given tensor. Then, a Bayesian dictionary learning (DL) model with tensor tubal transformed factorization, aiming to identify the underlying low-tubal-rank structure of the tensor effectively via the data-adaptive dictionary, is developed to solve the tensor robust principal component analysis (TRPCA) problem. With the defined pagewise tensor operators, a variational Bayesian DL algorithm is established and updates the posterior distributions instantaneously along the third dimension to solve the TPRCA. Extensive experiments on real-world applications, such as color image and hyperspectral image denoising and background/foreground separation problems, demonstrate both effectiveness and efficiency of the proposed approach in terms of various standard metrics.

Hyperspectral Anomaly Detection Based on Adaptive Low-Rank Transformed Tensor

Hyperspectral anomaly detection, which is aimed at distinguishing anomaly pixels from the surroundings in spatial features and spectral characteristics, has attracted considerable attention due to its various applications. In this article, we propose a novel hyperspectral anomaly detection algorithm based on adaptive low-rank transform, in which the input hyperspectral image (HSI) is divided into a background tensor, an anomaly tensor, and a noise tensor. To take full advantage of the spatial-spectral information, the background tensor is represented as the product of a transformed tensor and a low-rank matrix. The low-rank constraint is imposed on frontal slices of the transformed tensor to depict the spatial-spectral correlation of the HSI background. Besides, we initialize a matrix with predefined size and then minimize its l2.1 -norm to adaptively derive an appropriate low-rank matrix. The anomaly tensor is constrained with the l2.1.1 -norm to depict the group sparsity of anomalous pixels. We integrate all regularization terms and a fidelity term into a non-convex problem and develop a proximal alternating minimization (PAM) algorithm to solve it. Interestingly, the sequence generated by the PAM algorithm is proven to converge to a critical point. Experimental results conducted on four widely used datasets demonstrate the superiority of the proposed anomaly detector over several state-of-the-art methods.

Multiscale Feature Tensor Train Rank Minimization for Multidimensional Image Recovery

The general tensor-based methods can recover missing values of multidimensional images by exploiting the low-rankness on the pixel level. However, especially when considerable pixels of an image are missing, the low-rankness is not reliable on the pixel level, resulting in some details losing in their results, which hinders the performance of subsequent image applications (e.g., image recognition and segmentation). In this article, we suggest a novel multiscale feature (MSF) tensorization by exploiting the MSFs of multidimensional images, which not only helps to recover the missing values on a higher level, that is, the feature level but also benefits subsequent image applications. By exploiting the low-rankness of the resulting MSF tensor constructed by the new tensorization, we propose the convex and nonconvex MSF tensor train rank minimization (MSF-TT) to conjointly recover the MSF tensor and the corresponding original tensor in a unified framework. We develop the alternating directional method of multipliers (ADMMs) to solve the convex MSF-TT and the proximal alternating minimization (PAM) to solve the nonconvex MSF-TT. Moreover, we establish the theoretical guarantee of convergence for the PAM algorithm. Numerical examples of real-world multidimensional images show that the proposed MSF-TT outperforms other compared approaches in image recovery and the recovered MSF tensor can benefit the subsequent image recognition.

Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery

Recently, transform-based tensor nuclear norm (TNN) minimization methods have received increasing attention for recovering third-order tensors in multi-dimensional imaging problems. The main idea of these methods is to perform the linear transform along the third mode of third-order tensors and then minimize the nuclear norm of frontal slices of the transformed tensor. The main aim of this paper is to propose a nonlinear multilayer neural network to learn a nonlinear transform by solely using the observed tensor in a self-supervised manner. The proposed network makes use of the low-rank representation of the transformed tensor and data-fitting between the observed tensor and the reconstructed tensor to learn the nonlinear transform. Extensive experimental results on different data and different tasks including tensor completion, background subtraction, robust tensor completion, and snapshot compressive imaging demonstrate the superior performance of the proposed method over state-of-the-art methods.

Tensor Completion via Complementary Global, Local, and Nonlocal Priors

Completing missing entries in multidimensional visual data is a typical ill-posed problem that requires appropriate exploitation of prior information of the underlying data. Commonly used priors can be roughly categorized into three classes: global tensor low-rankness, local properties, and nonlocal self-similarity (NSS); most existing works utilize one or two of them to implement completion. Naturally, there arises an interesting question: can one concurrently make use of multiple priors in a unified way, such that they can collaborate with each other to achieve better performance? This work gives a positive answer by formulating a novel tensor completion framework which can simultaneously take advantage of the global-local-nonlocal priors. In the proposed framework, the tensor train (TT) rank is adopted to characterize the global correlation; meanwhile, two Plug-and-Play (PnP) denoisers, including a convolutional neural network (CNN) denoiser and the color block-matching and 3 D filtering (CBM3D) denoiser, are incorporated to preserve local details and exploit NSS, respectively. Then, we design a proximal alternating minimization algorithm to efficiently solve this model under the PnP framework. Under mild conditions, we establish the convergence guarantee of the proposed algorithm. Extensive experiments show that these priors organically benefit from each other to achieve state-of-the-art performance both quantitatively and qualitatively.