Spatio-temporal traffic data imputation is a fundamental component in intelligent transportation systems, which can significantly improve data quality and enhance the accuracy of downstream data mining tasks. Recently, low-rank tensor representation has shown great potential for spatio-temporal traffic data imputation. However, the low-rank assumption focuses on the global structure, neglecting the critical spatial topology and local temporal dependencies inherent in spatio-temporal data. To address these issues, we propose a topology-induced low-rank tensor representation (TILR), which can accurately capture the underlying low-rankness of the spatial multi-scale features induced by topology knowledge. Moreover, to exploit local temporal dependencies, we suggest a learnable convolutional regularization framework, which not only includes some classical convolution-based regularizers but also leads to the discovery of new convolutional regularizers. Equipped with the suggested TILR and convolutional regularizer, we build a unified low-rank tensor model harmonizing spatial topology and temporal dependencies for traffic data imputation, which is expected to deliver promising performance even under extreme and complex missing scenarios. To solve the proposed nonconvex model, we develop an efficient alternating direction method of multipliers (ADMM)-based algorithm and analyze its computational complexity. Extensive experiments demonstrate that the proposed model outperforms state-of-the-art baselines for various missing scenarios. These results reveal the critical synergy between topology-aware low-rank constraint and temporal dynamic modeling for spatio-temporal data imputation.

Remote sensing image (RSI) denoising is an important and fundamental task in RSI processing. Existing denoising methods usually assume that RSI lies in a single matrix or tensor subspace. However, due to the wavelength difference or/and temporal variability, the assumption of a single subspace may not be suitable for RSI. To address this, we propose a tensor multi-subspace representation (TenMSR) for RSI mixed noise removal. To be specific, in this work, we introduce TenMSR to finely characterize the intrinsic tensor multi-subspace structure of RSI. Compared with the single matrix/tensor subspace-based methods, the proposed method can not only precisely describe the wavelength difference or/and temporal variability of RSI but also produce a more compact image distribution in tensor multi-subspace. To mine and preserve the multi-subspace structure, we introduce a nonlinear transform-based 3-D tensor nuclear norm to characterize the tensor low rankness of the multi-subspace representation coefficient. An effective algorithm based on the proximal alternating minimization (PAM) framework is developed to solve the proposed model with theoretical convergence analysis. Extensive experiments show the effectiveness and superiority of the proposed method over existing state-of-the-art single matrix/tensor subspace RSI denoising methods.

Abstract. In this work, we study the subspace clustering and recovery of multidimensional images. Existing matrix-based/tensor-based subspace clustering methods successfully consider unilateral information (i.e., the similarity between image samples) to cluster samples into subspaces by using low-rank representation. The key issue of the unilateral representation-based methods is that the number of samples in each subspace should be sufficient for subspace representation. In practice, the clustering performance can be degraded when there is only a small number of observed samples in each subspace. To address the problem of insufficient observed samples, we propose to introduce hidden tensor data to supplement an insufficient number of observed samples. We employ both observed samples and hidden tensor data under low-rank constraints so that a new bilateral tensor low-rank representation (BTLRR) in subspace clustering is formulated. We show that a closed-form solution of block-diagonal tensor structure is obtained in subspace clustering of observed samples and hidden tensor data. Also the proposed BTLRR optimization problem can be solved by using the convex relaxation technique and augmented Lagrangian multiplier algorithm. The proposed BTLRR can fully explore the bilateral information of observations, including not only the similarity between samples but also the relationship among features. Extensive numerical results on multidimensional image data clustering and recovery illustrate that the effectiveness and robustness of the proposed bilateral representation are better than those of state-of-the-art methods (e.g., the popular LRR and TLRR methods).

Multi-view subspace clustering (MVSC) separates the data with multiple views into multiple clusters, and each cluster corresponds to one certain subspace. Existing tensor-based MVSC methods construct self-representation subspace coefficient matrices of all views as a tensor, and introduce the tensor nuclear norm (TNN) to capture the complementary information hidden in different views. The key assumption is that the data samples of each subspace must be sufficient for subspace representation. This work proposes a nonconvex latent transformed low-rank tensor representation framework for MVSC. To deal with the insufficient sample problem, we study the latent low-rank representation in the multi-view case to supplement underlying observed samples. Moreover, we propose to use data-driven transformed TNN (TTNN), resulting from the intrinsic structure of multi-view samples, to preserve the consensus and complementary information in the transformed domain. Meanwhile, the proposed unified nonconvex low-rank tensor representation framework can better learn the high correlation among different views. To resolve the proposed nonconvex optimization model, we propose an effective algorithm under the framework of the alternating direction method of multipliers and theoretically prove that the iteration sequences converge to the critical point. Experiments on various datasets showcase outstanding performance.

Abstract. This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the nonuniform one up to logarithmic factors, indicating that the uniformity costs very little. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for nonuniform recovery. These results complement the nonuniform ones recently obtained in Sun, Cui, and Liu [ IEEE Trans. Signal Process., 70 (2022), pp. 600–615] and provide more insights for understanding actual applications where the sensing ensemble is typically fixed and the corruption may be adversarial. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. We present experimental results to corroborate our theories. From the technical side, our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of a (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves the inner product. As a by-product, our QPE result refines the one in Xu and Jacques [ Inf. Inference, 9 (2020), pp. 543–586] under the sub-Gaussian random matrix, and in this specific instance, we are able to sharpen the uniform error decaying rate (for the projected back-projection estimator with signals in some convex symmetric set) presented therein from [Formula: see text] to [Formula: see text].

Recently, low-rank tensor completion (LRTC) methods by exploiting the global low-rankness of the target tensor have shown great potential for traffic data imputation. However, in real-world transportation networks, traffic data usually suffer from more complicated structural missing patterns than random-missing patterns, e.g., tube-missing patterns due to disruptions in wireless connections or slice-missing mechanism caused by sensor maintenance. As the naturally low-rank structure of traffic data in several missing scenarios, the existing LRTC methods indeed refrain from desirable performance for imputing traffic data. To tackle the complicated missing scenarios, we propose a convolutional low-rank tensor representation (CLRTR). Especially, CLRTR represents each unfolding matrix of the tensor as a sum of convolutions between two-dimensional (2D) filters and the corresponding low-rank coefficients, which allows us to simultaneously reveal the local patterns and the low-rankness of traffic data. Based on the CLRTR, we introduce the corresponding low-rank metric CLRTR-rank. Based on the suggested low-rank metric, we propose a traffic data imputation model that is well-suited to the complicated missing data scenarios. To implement the resultant imputation model, we design the alternating direction method of multipliers (ADMM) based algorithm with a theoretical convergence guarantee. Extensive numerical experiments on several real-world traffic datasets for both traffic data imputation and downstream traffic data prediction highlight the superiority of our model over the existing state-of-the-art matrix/tensor models for extensive missing scenarios.

Tensor analysis has received widespread attention in high-dimensional data learning. Unfortunately, the tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. How to recover the characteristics of the corrupted tensor data and make it compatible with the downstream clustering task remains a challenging problem. In this article, we study a generalized transformed tensor low-rank representation (TTLRR) model for simultaneously recovering and clustering the corrupted tensor data. The core idea is to find the latent low-rank tensor structure from the corrupted measurements using the transformed tensor singular value decomposition (SVD). Theoretically, we prove that TTLRR can recover the clean tensor data with a high probability guarantee under mild conditions. Furthermore, by using the transform adaptively learning from the data itself, the proposed TTLRR model can approximately represent and exploit the intrinsic subspace and seek out the cluster structure of the tensor data precisely. An effective algorithm is designed to solve the proposed model under the alternating direction method of multipliers (ADMMs) algorithm framework. The effectiveness and superiority of the proposed method against the compared methods are showcased over different tasks, including video/face data recovery and face/object/scene data clustering.

Recently, due to the ability of exploiting perceptual information, superpixel-based methods have received attention for hyperspectral image (HSI) denoising. However, existing superpixel-based denoising methods unfold the irregular 3D superpixels into the matrices along the spectral mode, which inevitably destroys the intrinsic structure of the irregular 3D superpixels. To tackle the irregular 3D superpixels, we introduce the irregular tensor representation for superpixels-guided HSI denoising. More concretely, by introducing the weighted tensor, we suggest a tensor representation of each irregular 3D superpixel, which can preserve the intrinsic structure of the irregular 3D superpixel. Equipped with the irregular tensor representation, we establish a superpixel-guided tensor optimization model for HSI denoising, which simultaneously exploits the perceptual information and low-rank structure within 3D superpixels. To solve the resulting tensor optimization problem, we develop an inexact augmented Lagrange multiplier (IALM) algorithm. Experimental results show that the proposed method outperforms other state-of-the-art HSI denoising methods, particularly matrix-based methods, on both simulated and real data.

Recently, block-term decomposition with rank-(Lr,Lr,1) (termed as LL1 decomposition), which is physically inspired by linear spectral unmixing, has received increasing attention in hyperspectral images (HSIs) denoising. However, due to the intrinsic nonlinear structure of real-world HSIs, the low-rankness of HSIs is usually implicit. Moreover, the essential uniqueness guarantee is usually violated with the low-rank assumption of the abundance maps unsupported in real scenarios, which hampers the successful deployment of LL1 decomposition. Inspired by the nonlinear spectral unmixing, we propose a nonlinear learnable transform-based LL1 decomposition (NT-LL1) for characterizing the implicit low-rank structure of real-world HSIs. More concretely, the nonlinear learnable transform in NT-LL1 decomposition is a composed transform consisting of a linear semi-orthogonal transform and a component-wise nonlinear transform, which collaboratively enhances the low-rankness of the abundance maps. Empowering with the NT-LL1 decomposition, we propose an NT-LL1 decomposition-based model for HSIs denoising. To tackle the resulting model, we develop an efficient proximal alternating minimization-based algorithm with a convergence guarantee. Extensive experimental results including simulated and real data collectively verify the superiority of the proposed method as compared with the competing methods.

Multi-view clustering (MVC) aims at exploiting the consistent features within different views to divide samples into different clusters. Existing subspace-based MVC algorithms usually assume linear subspace structures and two-stage similarity matrix construction strategies, thereby posing challenges in imprecise low-dimensional subspace representation and inadequacy of exploring consistency. This paper presents a novel hierarchical representation for MVC method via the integration of intra-sample, intra-view, and inter-view representation learning models. In particular, we first adopt the deep autoencoder to adaptively map the original high-dimensional data into the latent low-dimensional representation of each sample. Second, we use the self-expression of the latent representation to explore the global similarity between samples of each view and obtain the subspace representation coefficients. Third, we construct the third-order tensor by arranging multiple subspace representation matrices and impose the tensor low-rank constraint to sufficiently explore the consistency among views. Being incorporated into a unified framework, these three models boost each other to achieve a satisfactory clustering result. Moreover, an alternating direction method of multipliers algorithm is developed to solve the challenging optimization problem. Extensive experiments on both simulated and real-world multi-view datasets show the superiority of the proposed method over eight state-of-the-art baselines.

The hyperspectral unmixing (HU) aims at estimating the spectral signatures of endmembers (or materials) and their corresponding abundance maps of the hyperspectral image (HSI). In this work, we treat the HSI as an 3D cube and propose a new tensor-based HU method. We decompose an HSI data as the sum of several multilinear rank-(Lr, Lr, 1) terms (or LL1 model). Based on the fact that the latent factors of LL1 model are physical meaningful (i.e., abundance maps and spectral signatures), we build a nonnegative tensor decomposition optimization model with the low-rank constraint and impose an implicit regularizer to exploit the nonlocal self-similarity prior of abundance maps-whose related subproblem can be easily solved under the plug-and-play framework. We develop an alternating direction method of multipliers algorithm to solve the proposed model. Numerical experiments demonstrate the effectiveness of our algorithm.

For multi-temporal remote sensing (RS) images, the distribution of surface materials is constant concerning time and the same material shows different spectral features at different times. Decomposing the image at each time into an abundance tensor and temporal features, there is a strong similarity between abundance tensors of all time. Based on this observation, we suggest a blind thick cloud/shadow removal model, which exploits the sparsity of the cloud component and the similarity between abundance tensors, achieving both cloud detection and multi-temporal information restoration. Moreover, a mask refinement strategy is designed to pursue the optimal cloud/shadow mask. We develop an efficient algorithm to solve the proposed model based on the augmented Lagrange multiplier method. The results of simulated experiments in different scenarios verify the superiority of the proposed method for thick cloud/shadow removal.