Low-rank Tensor Research Articles

A novel data-driven 3D site characterisation method: Tucker decomposition-Bayesian compressive sensing and benchmarking study

ABSTRACT Site characterisation plays a pivotal role in geotechnical design and analysis. With advancements in machine learning and other digital technologies, data-driven site characterisation (DDSC) has garnered substantial interest in data-centric geotechnics. This paper proposes a novel and competitive 3D DDSC method, called Tucker decomposition-Bayesian compressive sensing (TD-BCS), which employs Tucker decomposition to factorise a 3D high-order tensor into a low-rank core tensor along with three associated factor matrices. The method comprises three key components: (1) 3D sparse representation using Tucker decomposition and discrete cosine transform; (2) Bayesian compressive sensing, and its algorithm implementation; and (3) Determination of three effective factor matrices. The proposed TD-BCS method is evaluated through its application to two benchmark examples, involving virtual ground and actual ground scenario based on real CPT data. And the benchmarking results demonstrate that the method not only yields accurate estimates with quantified uncertainty from sparse measurements but also exhibits high computational efficiency compared to other DDSC methods. Indeed, the TD-BCS method can be effectively applied to other 3D geotechnical engineering cases, and its framework is readily extendable to higher dimensions.

A low-rank support tensor machine for multi-classification

In recent decades, there has been an increasing demand for effectively handling high-dimensional multi-channel tensor data. Due to the inability to utilize internal structural information, Support Vector Machine (SVM) and its variations struggle to classify flattened tensor data, consequently resulting in the ‘curse of dimensionality’ issue. Furthermore, most of these methods can not directly apply to multiclass datasets. To overcome these challenges, we have developed a novel classification method called Multiclass Low-Rank Support Tensor Machine (MLRSTM). Our method is inspired by the well-established low-rank tensor hypothesis, which suggests a correlation between each channel of the feature tensor. Specifically, MLRSTM adopts the hinge loss function and introduces a convex approximation of tensor rank, the order-d Tensor Nuclear Norm (order-d TNN), in the regularization term. By leveraging the order-d TNN, MLRSTM effectively exploits the inherent structural information in tensor data to enhance generalization performance and avoid the curse of dimensionality. Moreover, we develop the Alternating Direction Method of Multipliers (ADMM) algorithm to optimize the convex problem inherent in training MLRSTM. Finally, comprehensive experiments validate the excellent performance of MLRSTM in tensor multi-classification tasks, showcasing its potential and efficacy in handling high-dimensional multi-channel tensor data.

Joint local smoothness and low-rank tensor representation for robust multi-view clustering

Low-rank tensor representation (LRTR) has become a significant method for achieving improved multi-view clustering (MVC) performance. Generally, most LRTR methods impose a tensor low-rank constraint (TLRC) on a tensor, which is spliced by the representation matrix of each view, to explore the low-rank prior hidden in these representation matrices. However, two problems remain unsolved. On the one hand, these representation matrices still contain noise since the raw data usually possess noise; on the other hand, and more importantly, is there any other prior information that can be explored among these representation matrices? Since samples located in the same cluster are similar, the coefficients of their representation matrices should exhibit similar. That is, these representation matrices should be local smoothness (LS), which is also verified by our numerical tests. In addition, the use of the LS prior helps to denoise the noise contain in the data. Then, in this paper, we propose a joint LS and LRTR for robust MVC method (LS-LRTR), which can simultaneously exploit the low-rank and LS priors. Specifically, we utilize a TLRC to explore the low-rank prior in the representation matrices. Subsequently, to mine the LS prior and further reduce the influence of noise, we introduce the Total Variation (TV) norm to the constraint representation matrices. Then, we fuse the TLRC and TV norm into a unified framework. Additionally, we apply an Augmented Lagrange Multiplier to solve the optimization problem of LS-LRTR. Experiments conducted on several datasets indicate that LS-LRTR outperforms the state-of-the-art clustering methods.

A modified block Hessenberg method for low-rank tensor Sylvester equation

This work focuses on iteratively solving the tensor Sylvester equation with low-rank right-hand sides. To solve such equations, we first introduce a modified version of the block Hessenberg process so that approximation subspaces contain some extra block information obtained by multiplying the initial block by the inverse of each coefficient matrix of the tensor Sylvester equation. Then, we apply a Galerkin-like condition to transform the original tensor Sylvester equation into a low-dimensional tensor form. The reduced problem is then solved using a blocked recursive algorithm based on Schur decomposition. Moreover, we reveal how to stop the iterations without the need to compute the approximate solution by calculating the residual norm or an upper bound. Eventually, some numerical examples are given to assess the efficiency and robustness of the suggested method.

Boundary points guided 3D object detection for point clouds

3D object detection in LiDAR point clouds is crucial for computer vision tasks such as autonomous driving. The two-stage approach with point cloud completion achieves remarkable performance by generating semantic surface points for foreground objects or learning bird’s eye view shape heat map labels. However, these methods require additional completion datasets, leading to substantial computation and memory demands. In this context, we propose a Boundary Points Guided 3D (BPG3D) object detection method that complements point cloud boundary information without the need for additional data. Specifically, we generate Region of Interest (RoI) boundary points to aggregate the neighbor voxel information at the RoI boundary during the refinement stage to complement the missing boundary information. Meanwhile, we design a Dual Feature Selection (DFS) module to adaptively fuse RoI grid point features and RoI boundary point features for bounding box refinement with negligible computational cost. Additionally, inspired by tensor decomposition theory, we use low-rank tensors to reconstruct high-rank tensors in the point cloud feature encoder to enhance contextual semantic information. The proposed method achieves 65.81% mAP on KITTI Test Set, obtaining a good trade-off between accuracy and efficiency.

Ferumoxytol-Enhanced 5D Multiphase Steady-State Imaging Using Rotating Cartesian K-Space With Low-Rank Reconstruction for Pediatric Congenital Heart Disease.

The rotating Cartesian k-space multiphase steady-state imaging with contrast (ROCK-MUSIC) pulse sequence enables acquisition of whole-heart, cardiac phase-resolved images in pediatric congenital heart disease (CHD) without reliance on the ventilator gating signal. Multidimensional reconstruction with low rank tensor (LRT) has shown promise for resolving complex cardiorespiratory motion. To enhance ROCK-MUSIC by resolving cardiorespiratory phases using LRT reconstruction and to enable semi-automatic hyperparameter tuning by developing an image quality scoring model. Retrospective. Thirty patients (45% female, age 2 days to 6.7 years) with CHD. 3-T, four-dimensional (4D) spoiled gradient recalled echo sequence. Eigenvector-based iTerative Self-consistent Parallel Imaging Reconstruction (ESPIRiT) served as the reference comparison for LRT reconstruction. A 4-point Likert scale was used for cardiac and vascular image quality scoring based on cardiac chamber definition, lumen signal uniformity, vascular margin clarity, and motion artifact. Ejection fraction and ventricular volumes were assessed in 16 patients. Signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and edge sharpness were computed. Intraclass correlation coefficients, Wilcoxon signed-rank test, Bland-Altman. A P-value <0.05 was considered statistically significant. Relative to ESPIRiT, LRT images received significantly higher cardiac (2.81 ± 0.57 vs. 3.19 ± 0.54) and vascular (2.81 ± 0.60 vs. 3.36 ± 0.53) image quality scores. Image quality scoring with semi-automated hyperparameter tuning showed strong correlations (R2 = 0.748) among image quality, SNR, and septal sharpness. Comparison of ejection fraction and volumetry derived from ESPIRiT, and LRT showed no significant systematic difference (P = 0.32). Integration of low-rank reconstruction with ROCK-MUSIC acquisition may be feasible, and semi-automatic hyperparameter tuning could be effective for generating cardiorespiratory resolved images. 2 TECHNICAL EFFICACY: Stage 1.

High Spatial-Resolution and Acquisition-Efficiency Cardiac MR T1 Mapping Based on Radial bSSFP and a Low-Rank Tensor Constraint.

Cardiac T1 mapping is valuable for evaluating myocardial fibrosis, yet its resolution and acquisition efficiency are limited, potentially obscuring visualization of small pathologies. To develop a technique for high-resolution cardiac T1 mapping with a less-than-100-millisecond acquisition window based on radial MOdified Look-Locker Inversion recovery (MOLLI) and a calibrationless space-contrast-coil locally low-rank tensor (SCC-LLRT) constrained reconstruction. Prospective. Sixteen healthy subjects (age 25 ± 3 years, 44% females) and 12 patients with suspected cardiomyopathy (age 57 ± 15 years, 42% females), NiCl2-agar phantom. 3-T, standard MOLLI, radial MOLLI, inversion-recovery spin-echo, late gadolinium enhancement. SCC-LLRT was compared to a conventional locally low-rank (LLR) method through simulations using Normalized Root-Mean-Square Error (NRMSE) and Structural Similarity Index Measure (SSIM). Radial MOLLI was compared to standard MOLLI across phantom, healthy subjects, and patients. Three independent readers subjectively evaluated the quality of T1 maps using a 5-point scale (5 = best). Paired t-test, Wilcoxon signed-rank test, intraclass correlation coefficient analysis, linear regression, Bland-Altman analysis. P < 0.05 was considered statistically significant. In simulations, SCC-LLRT demonstrated a significant improvement in NRMSE and SSIM compared to LLR. In phantom, both radial MOLLI and standard MOLLI provided consistent T1 estimates across different heart rates. In healthy subjects, radial MOLLI exhibited a significantly lower mean T1 (1115 ± 39 msec vs. 1155 ± 36 msec), similar T1 SD (74 ± 14 msec vs. 67 ± 23 msec, P = 0.20), and similar T1 reproducibility (28 ± 18 msec vs. 22 ± 15 msec, P = 0.34) compared to standard MOLLI. In patients, the proposed method significantly improved the sharpness of myocardial boundaries (4.50 ± 0.65 vs. 3.25 ± 0.43), the conspicuity of papillary muscles and fine structures (4.33 ± 0.74 vs. 3.33 ± 0.47), and artifacts (4.75 ± 0.43 vs. 3.83 ± 0.55). The reconstruction time for a single slice was 5.2 hours. The proposed method enables high-resolution cardiac T1 mapping with a short acquisition window and improved image quality. 1 TECHNICAL EFFICACY: Stage 1.

Low-rank tensor completion based on tensor train rank with partially overlapped sub-blocks and total variation

Recently, the low-rank tensor completion method based on tensor train (TT) rank has achieved promising performance. Ket augmentation (KA) is commonly used in TT rank-based methods to improve the performance by converting low-dimensional tensors to higher-dimensional tensors. However, block artifacts are caused since KA also destroys the original structure and image continuity of original low-dimensional tensors. To tackle this issue, a low-rank tensor completion method based on TT rank with tensor augmentation by partially overlapped sub-blocks (TAPOS) and total variation (TV) is proposed in this paper. The proposed TAPOS preserves the image continuity of the original tensor and enhances the low-rankness of the generated higher-dimensional tensors, and a weighted de-augmentation method is used to assign different weights to the elements of sub-blocks and further reduce the block artifacts. To further alleviate the block artifacts and improve reconstruction accuracy, TV is introduced in the TAPOS-based model to add the piecewise smooth prior. The parallel matrix decomposition method is introduced to estimate the TT rank to reduce the computational cost. Numerical experiments show that the proposed method outperforms the existing state-of-the-art methods.

Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems.

Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.

Tensor approximation of functional differential equations.

Functional differential equations(FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations(PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance (parallel) tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equationevolving from a random initial state.

Cyclic tensor singular value decomposition with applications in low-rank high-order tensor recovery

The rapid advancements in emerging technologies have increased the demand for recovery tasks involving high-dimensional data with complex structures. Effectively utilizing tensor decomposition techniques to capture the low-rank structure of such data is crucial. Recently, the high-order t-SVD has demonstrated strong adaptability. However, this decomposition approach can only capture the low-rank correlation of two modes along other modes individually, while disregarding the structural correlation between different modes. In this paper, we propose a novel cyclic tensor singular value decomposition (CTSVD) method that effectively characterizes the low-rank structures of high-order tensors along all modes. Specifically, our method decomposes an order-N tensor into N factor tensors and one core tensor, connecting them using a defined mode-k tensor-tensor product (t-product). Building upon this, we establish the corresponding tensor rank and its convex relaxation. To address the issue of dimensional imbalance between adjacent modes in high-dimensional data, we propose and integrate a square reshaping strategy into the recovery models for tensor completion (TC) and tensor principal component analysis (TRPCA) tasks. Effective alternating direction method of multipliers (ADMM)-based algorithms are designed to these tasks. Extensive experiments on both synthetic and real data demonstrate that our methods outperform state-of-the-art approaches.

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

A novel ship trajectory reconstruction approach based on low-rank tensor completion

The Automatic Identification System (AIS) enhances maritime safety and environmental monitoring by providing crucial ship trajectory data. However, this data is often compromised by missing or abnormal values due to signal blockage, transmission errors, and equipment failures, jeopardizing safety and the accuracy of environmental analyses. Addressing these challenges, we propose a novel Customized Tensor-Based Maritime Trajectory Reconstruction (CTMTR) framework that leverages the self-similarity of ship trajectories to reformulate trajectory reconstruction as a matrix rank minimization issue. The CTMTR framework consists of three steps: data preprocessing, trajectory matrix construction, and trajectory reconstruction. We conducted simulation experiments using a dataset comprising vessel trajectories from the east coast of Dover in the United States in 2022. To validate its effectiveness, the CTMTR is compared with four advanced methods (LRMC, LRTC-TNN, TRPCA, and TNN-DCT) under diverse missing and anomalous scenarios. The results substantiate that the performance of CTMTR outperforms other approaches, especially in anomalous cases. Our method outperforms existing methods by an order of magnitude under random missing combined with anomaly scenarios. The CTMTR framework thus has the potential to advance maritime trajectory reconstruction methodologies, providing a solid foundation for future innovations in maritime data analysis and navigation safety technologies.

TIHN: Tensor Improved Huber Norm for low-rank tensor recovery

Tensor Robust Principal Component Analysis (TRPCA) has received much attention in many real-world applications which aims to recover a low-rank tensor corrupted by sparse noise. Most existing TRPCA methods usually regularize the low-rank component by minimizing its Tensor Nuclear Norm (TNN). However, the original TNN shrinks all of the singular values with the same penalty which generally leads to suboptimal results. The reason is that the large singular values should be less penalized since they usually correspond to salient information of the real-world tensor. In this work, we develop a Tensor Improved Huber Norm (TIHN) which considers the difference between the singular values of tensor. The TIHN can achieve better performance by recovering the large and significant singular values exactly. We also establish the Huber TRPCA (HTRPCA) method by utilizing the proposed TIHN. In addition, an efficient optimization algorithm based on the half-quadratic theory and Alternating Direction Method of Multipliers (ADMM) framework is designed to implement the HTRPCA. Finally, experiments on color image recovery and video recovery validate the effectiveness of the proposed method.

Advancements in Remote Compressive Hyperspectral Imaging: Adaptive Sampling with Low-Rank Tensor Image Reconstruction

We advanced the practical development of compressive hyperspectral cameras for remote sensing scenarios with a design that simultaneously compresses and captures high-quality spectral information of a scene via configurable measurements. We built a prototype imaging system that is compatible with light-modulation devices that encode the incoming spectrum. The sensing approach enables a substantial reduction in the volume of data collected and transmitted, facilitating large-scale remote hyperspectral imaging. A main advantage of our sensing design is that it allows for adaptive sampling. When prior information of a survey region is available or gained, the modulation patterns can be re-programmed to efficiently sample and detect desired endmembers. Given target spectral signatures, we propose an optimization scheme that guides the encoding process. The approach severely reduces the number of required sampling patterns, with the ability to achieve image segmentation and correct distortions. Additionally, to decode the modulated data, we considered a novel reconstruction algorithm suited for large-scale images. The computational methodology leverages the multidimensional structure and redundant representation of hyperspectral images via the canonical polyadic decomposition of multiway arrays. Under realistic remote sensing scenarios, we demonstrated the efficiency of our approach with several data sets collected by our prototype camera and reconstructed by our low-rank tensor decoder.

Tensorized Incomplete Multi-view Kernel Subspace Clustering

Recently considerable advances have been achieved in the incomplete multi-view clustering (IMC) research. However, the current IMC works are often faced with three challenging issues. First, they mostly lack the ability to recover the nonlinear subspace structures in the multiple kernel spaces. Second, they usually neglect the high-order relationship in multiple representations. Third, they often have two or even more hyper-parameters and may not be practical for some real-world applications. To tackle these issues, we present a Tensorized Incomplete Multi-view Kernel Subspace Clustering (TIMKSC) approach. Specifically, by incorporating the kernel learning technique into an incomplete subspace clustering framework, our approach can robustly explore the latent subspace structure hidden in multiple views. Furthermore, we impute the incomplete kernel matrices and learn the low-rank tensor representations in a mutual enhancement manner. Notably, our approach can discover the underlying relationship among the observed and missing samples while capturing the high-order correlation to assist subspace clustering. To solve the proposed optimization model, we design a three-step algorithm to efficiently minimize the unified objective function, which only involves one hyper-parameter that requires tuning. Experiments on various benchmark datasets demonstrate the superiority of our approach. The source code and datasets are available at: https://www.researchgate.net/publication/381828300_TIMKSC_20240629.

Efficient enhancement of low-rank tensor completion via thin QR decomposition.

Low-rank tensor completion (LRTC), which aims to complete missing entries from tensors with partially observed terms by utilizing the low-rank structure of tensors, has been widely used in various real-world issues. The core tensor nuclear norm minimization (CTNM) method based on Tucker decomposition is one of common LRTC methods. However, the CTNM methods based on Tucker decomposition often have a large computing cost due to the fact that the general factor matrix solving technique involves multiple singular value decompositions (SVDs) in each loop. To address this problem, this article enhances the method and proposes an effective CTNM method based on thin QR decomposition (CTNM-QR) with lower computing complexity. The proposed method extends the CTNM by introducing tensor versions of the auxiliary variables instead of matrices, while using the thin QR decomposition to solve the factor matrix rather than the SVD, which can save the computational complexity and improve the tensor completion accuracy. In addition, the CTNM-QR method's convergence and complexity are analyzed further. Numerous experiments in synthetic data, real color images, and brain MRI data at different missing rates demonstrate that the proposed method not only outperforms in terms of completion accuracy and visualization, but also conducts more efficiently than most state-of-the-art LRTC methods.

Joint low-rank tensor fusion and cross-modal attention for multimodal physiological signals based emotion recognition

Objective. Physiological signals based emotion recognition is a prominent research domain in the field of human-computer interaction. Previous studies predominantly focused on unimodal data, giving limited attention to the interplay among multiple modalities. Within the scope of multimodal emotion recognition, integrating the information from diverse modalities and leveraging the complementary information are the two essential issues to obtain the robust representations. Approach. Thus, we propose a intermediate fusion strategy for combining low-rank tensor fusion with the cross-modal attention to enhance the fusion of electroencephalogram, electrooculogram, electromyography, and galvanic skin response. Firstly, handcrafted features from distinct modalities are individually fed to corresponding feature extractors to obtain latent features. Subsequently, low-rank tensor is fused to integrate the information by the modality interaction representation. Finally, a cross-modal attention module is employed to explore the potential relationships between the distinct latent features and modality interaction representation, and recalibrate the weights of different modalities. And the resultant representation is adopted for emotion recognition. Main results. Furthermore, to validate the effectiveness of the proposed method, we execute subject-independent experiments within the DEAP dataset. The proposed method has achieved the accuracies of 73.82% and 74.55% for valence and arousal classification. Significance. The results of extensive experiments verify the outstanding performance of the proposed method.

Low-Rank Tensor Completion Based on Self-Adaptive Learnable Transforms.

The tensor nuclear norm (TNN), defined as the sum of nuclear norms of frontal slices of the tensor in a frequency domain, has been found useful in solving low-rank tensor recovery problems. Existing TNN-based methods use either fixed or data-independent transformations, which may not be the optimal choices for the given tensors. As the consequence, these methods cannot exploit the potential low-rank structure of tensor data adaptively. In this article, we propose a framework called self-adaptive learnable transform (SALT) to learn a transformation matrix from the given tensor. Specifically, SALT aims to learn a lossless transformation that induces a lower average-rank tensor, where the Schatten- p quasi-norm is used as the rank proxy. Then, because SALT is less sensitive to the orientation, we generalize SALT to other dimensions of tensor (SALTS), namely, learning three self-adaptive transformation matrices simultaneously from given tensor. SALTS is able to adaptively exploit the potential low-rank structures in all directions. We provide a unified optimization framework based on alternating direction multiplier method for SALTS model and theoretically prove the weak convergence property of the proposed algorithm. Experimental results in hyperspectral image (HSI), color video, magnetic resonance imaging (MRI), and COIL-20 datasets show that SALTS is much more accurate in tensor completion than existing methods. The demo code can be found at https://faculty.uestc.edu.cn/gaobin/zh_CN/lwcg/153392/list/index.htm.

Robust Corrupted Data Recovery and Clustering via Generalized Transformed Tensor Low-Rank Representation.

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.