Tensor Completion Research Articles

Low-rank quaternion tensor completion for color video inpainting via a novel factorization strategy

Recently, a quaternion tensor product named Qt-product was proposed, and then the singular value decomposition and the rank of a third-order quaternion tensor were given. From a more applicable perspective, we extend the Qt-product and propose a novel multiplication principle for third-order quaternion tensor named gQt-product. With the gQt-product, we introduce a brand-new singular value decomposition for third-order quaternion tensors named gQt-SVD and then define gQt-rank and multi-gQt-rank. We prove that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos. So, we apply the low-rank quaternion tensor completion to color video inpainting problems and present alternating least-square algorithms to solve the proposed low gQt-rank and multi-gQt-rank quaternion tensor completion models. The convergence analyses of the proposed algorithms are established and some numerical experiments on various color video datasets show the high recovery accuracy and computational efficiency of our methods.

Learnable Transform-Assisted Tensor Decomposition for Spatio-Irregular Multidimensional Data Recovery

Tensor decompositions have been successfully applied to multidimensional data recovery. However, classical tensor decompositions are not suitable for emerging spatio-irregular multidimensional data (i.e., spatio-irregular tensor), whose spatial domain is non-rectangular, e.g., spatial transcriptomics data from bioinformatics and semantic units from computer vision. By using preprocessing (e.g., zero-padding or element-wise 0-1 weighting), the spatio-irregular tensor can be converted to a spatio-regular tensor and then classical tensor decompositions can be applied, but this strategy inevitably introduces bias information, leading to artifacts. How to design a tensor-based method suitable for emerging spatio-irregular tensors is an imperative challenge. To address this challenge, we propose a learnable transform-assisted tensor singular value decomposition (LTA-TSVD) for spatio-irregular tensor recovery, which allows us to leverage the intrinsic structure behind the spatio-irregular tensor. Specifically, we design a learnable transform to project the original spatio-irregular tensor into its latent spatio-regular tensor, and then the latent low-rank structure is captured by classical TSVD on the resulting regular tensor. Empowered by LTA-TSVD, we develop spatio-irregular low-rank tensor completion (SIR-LRTC) and spatio-irregular tensor robust principal component analysis (SIR-TRPCA) models for the spatio-irregular tensor imputation and denoising respectively, and we design corresponding solving algorithms with theoretical convergence. Extensive experiments including the spatial transcriptomics data imputation and hyperspectral image denoising show SIR-LRTC and SIR-TRPCA are superior performance to competing approaches and benefit downstream applications.

A tensor completion method based on tensor QR decomposition with truncated nuclear norm and sparse regularization

A tensor completion method based on tensor QR decomposition with truncated nuclear norm and sparse regularization

Adaptive sampling with tensor leverage scores for exact low-rank third-order tensor completion

Tensor completion aims at estimating the missing entries from the incomplete observation. Under the tensor singular value decomposition framework, the exact recovery of a low-tubal-rank third-order tensor could be achieved via convex optimization with high probability if the tensor satisfies the tensor incoherence condition. In this work, we show that, when the random selection of entries is made adaptive to a distribution which is dependent on the coherence structure of the tensor, any low-tubal-rank tensor of the size n×n×n with tubal-rank r can be exactly recovered with high probability from as few as O(rn2log2⁡(n)) randomly chosen entries. In practice, tensor leverage scores are not known a priori, and we design a two-phase adaptive sampling strategy to obtain the leverage scores. Numerical experiments on synthetic and real-world third-order tensor data sets are used to validate our theoretical results and illustrate that the tensor recovery performance of the proposed two-phase adaptive sampling scheme is better than that of the other state-of-the-art methods.

Self representation based methods for tensor completion problem

Tensor, the higher-order data array, naturally arises in many fields, such as information sciences, seismic data reconstruction, physics, video inpainting and so on. In this paper, we intend to provide a new model to recover a tensor, based on self-representation, for the all-mode unfoldings of the desired tensor, regardless of the tensor rank. The suggested idea generalizes self-representation to tensor and recovers an incomplete tensor by reconstructing one fiber by others in such a way that they all belong to the same subspace. We design least-square and low-rank self-representation algorithms based on the Linearized Alternating Direction Method utilizing this concept. We show that the proposed algorithms converge to the rank-minimization of the incomplete tensor.

CP decomposition-based algorithms for completion problem of motion capture data

Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset.

Laplacian-enhanced non-synchronous measurements method

To address the challenges of physical geometric constraints of microphone arrays and the lack of a prior information about the operating frequencies of target signals, we propose a novel method for broadband multiple-sound-source localization: a non-synchronous measurements method for broadband multiple-sound-source localization based on Laplacian-enhanced low-rank tensor completion. Our method thoroughly analyzes the tensor data structure of broadband signals and introduces a multiplier-optimized alternating direction method. Through extensive simulation studies, our approach has achieved remarkable results. We validate the outstanding performance of the Laplacian-enhanced algorithm, demonstrating its ability to generate highly accurate sound maps, capturing five distinct speech signal sources. Compared to other methods, our approach provides a more comprehensive global view. This technological advancement brings substantial hope for the precise localization of multiple broadband sound sources in critical applications, offering significant potential for future applications.

Robust Low-Tubal-Rank Tensor Completion Based on Tensor Factorization and Maximum Correntopy Criterion.

The goal of tensor completion is to recover a tensor from a subset of its entries, often by exploiting its low-rank property. Among several useful definitions of tensor rank, the low tubal rank was shown to give a valuable characterization of the inherent low-rank structure of a tensor. While some low-tubal-rank tensor completion algorithms with favorable performance have been recently proposed, these algorithms utilize second-order statistics to measure the error residual, which may not work well when the observed entries contain large outliers. In this article, we propose a new objective function for low-tubal-rank tensor completion, which uses correntropy as the error measure to mitigate the effect of the outliers. To efficiently optimize the proposed objective, we leverage a half-quadratic minimization technique whereby the optimization is transformed to a weighted low-tubal-rank tensor factorization problem. Subsequently, we propose two simple and efficient algorithms to obtain the solution and provide their convergence and complexity analysis. Numerical results using both synthetic and real data demonstrate the robust and superior performance of the proposed algorithms.

Tensor Completion Using Bilayer Multimode Low-Rank Prior and Total Variation.

In this article, we propose a novel bilayer low-rankness measure and two models based on it to recover a low-rank (LR) tensor. The global low rankness of underlying tensor is first encoded by LR matrix factorizations (MFs) to the all-mode matricizations, which can exploit multiorientational spectral low rankness. Presumably, the factor matrices of all-mode decomposition are LR, since local low-rankness property exists in within-mode correlation. In the decomposed subspace, to describe the refined local LR structures of factor/subspace, a new low-rankness insight of subspace: a double nuclear norm scheme is designed to explore the so-called second-layer low rankness. By simultaneously representing the bilayer low rankness of the all modes of the underlying tensor, the proposed methods aim to model multiorientational correlations for arbitrary N -way ( N ≥ 3 ) tensors. A block successive upper-bound minimization (BSUM) algorithm is designed to solve the optimization problem. Subsequence convergence of our algorithms can be established, and the iterates generated by our algorithms converge to the coordinatewise minimizers in some mild conditions. Experiments on several types of public datasets show that our algorithm can recover a variety of LR tensors from significantly fewer samples than its counterparts.

A novel weighted-guided tensor completion missing data imputation method for health monitoring data of planar parallel mechanism

Abstract High-end mechanical equipment plays a crucial role in the manufacturing industry, making the monitoring of its operational status highly significant. Due to various factors such as environmental influences, the absence of monitoring signals in mechanical equipment status is a common issue, leading to a decline in data quality. To ensure data quality, this paper proposes an adaptive weighted low-rank tensor missing data imputation method. Firstly, based on the motion characteristics of a planar parallel mechanism (PPM), a new low-rank tensor model is established using periodicity, sliding windows, and time series. Secondly, the tensor truncation nuclear norm is defined, and a novel rate parameter is introduced to control the truncation degree of all tensor modes, thereby obtaining weights for each dimension. Finally, within the framework of the alternating direction method of multipliers (ADMM), adaptive weights for each dimension are obtained during each iteration, completing the filling of missing data. Two types of missing patterns are studied on a PPM experimental platform, and the results showed that as the missing rate increased, the mean absolute percentage error is less than 0.018, and root mean square error is less than 0.001, and the rate of change is less than 0.08, which was significantly better than other compared algorithms.

Low-rank sparse fully-connected tensor network for tensor completion

Fully-connected tensor network (FCTN) has recently drawn lots of attention in tensor completion due to its full description of all correlations between any two modes. However, the FCTN model has multiple ranks, and existing methods often ignore the difficulty brought about by rank selection, especially when the model is rank-sensitive. To overcome this drawback, this paper proposed a low-rank sparse FCTN for tensor completion. Specifically, we theoretically show that, for a tensor with FCTN structure, its subtensor can be represented as a coefficient sum of basic tensors, where the coefficients are remained in the FCTN’s factors. This means that factors’ sparsity can improve the robustness of FCTN to larger rank selection. Moreover, according to the relation between the target tensor and its FCTN’s factors, low-rank constrain is used to enhance rank robustness. Lastly, we optimize the proposed model by the alternating direction method of multipliers (ADMM) algorithm. Experimental results show that the low-rank sparse constraint effectively improves the rank robustness of the FCTN model, and achieves excellent results compared with other state-of-the-art completion methods.

A robust low‐rank tensor completion model with sparse noise for higher‐order data recovery

AbstractThe tensor singular value decomposition‐based model has garnered increasing attention in addressing tensor recovery challenges. However, existing tensor recovery methods exhibit certain inherent limitations. Some ignore the simultaneous effects of noise and missing values, while most can't handle higher‐order tensors, which are not reflective of real‐world scenarios. The information redundancy within tensor data often leads to a prevailing low‐rank structure, making low‐rankness a vital prior in the tensor recovery process. To tackle this pressing issue, a robust low‐rank tensor recovery framework is proposed to rehabilitate higher‐order tensors corrupted by sparse noise and missing entries. In the model, the tensor nuclear norm derived for order‐d tensors (d 4) are employed as a representation of the low‐rank prior, while utilizing the ‐norm to model the sparse noise. To solve the proposed model, an efficient Alternating direction method of multipliers algorithm is developed. A series of experiments are performed on synthetic and real‐world datasets. The results show that the superior performance of the method compared with other algorithms dedicated to addressing order‐d tensor recovery challenges. Notably, in scenarios where the data is severely compromised (noise ratio 40%, sample ratio 70%), the algorithm consistently outperforms its competitors, achieving significantly improved results.

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.

Hyperspectral Tensor Completion Using Low-Rank Modeling and Convex Functional Analysis.

Hyperspectral tensor completion (HTC) for remote sensing, critical for advancing space exploration and other satellite imaging technologies, has drawn considerable attention from recent machine learning community. Hyperspectral image (HSI) contains a wide range of narrowly spaced spectral bands hence forming unique electrical magnetic signatures for distinct materials, and thus plays an irreplaceable role in remote material identification. Nevertheless, remotely acquired HSIs are of low data purity and quite often incompletely observed or corrupted during transmission. Therefore, completing the 3-D hyperspectral tensor, involving two spatial dimensions and one spectral dimension, is a crucial signal processing task for facilitating the subsequent applications. Benchmark HTC methods rely on either supervised learning or nonconvex optimization. As reported in recent machine learning literature, John ellipsoid (JE) in functional analysis is a fundamental topology for effective hyperspectral analysis. We therefore attempt to adopt this key topology in this work, but this induces a dilemma that the computation of JE requires the complete information of the entire HSI tensor that is, however, unavailable under the HTC problem setting. We resolve the dilemma, decouple HTC into convex subproblems ensuring computational efficiency, and show state-of-the-art HTC performances of our algorithm. We also demonstrate that our method has improved the subsequent land cover classification accuracy on the recovered hyperspectral tensor.

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.

A new disease mapping method for improving data completeness of syndromic surveillance with high missing rates

AbstractSyndromic surveillance is a type of public health surveillance that utilizes nonspecific indicators or symptoms associated with a particular disease or condition to detect and track disease outbreaks early. However, data completeness has been a significant challenge for syndromic surveillance systems in many countries. Incomplete data may make it difficult to accurately identify anomalies or trends in surveillance data. In this study, a new disease mapping method based on a high‐accuracy, low‐rank tensor completion (HaLRTC) algorithm is proposed to estimate the quarterly positivity rate of the human influenza virus (IFV) based on highly insufficient 2010–2015 respiratory syndromic surveillance data from the subtropical monsoon region of China. The HaLRTC algorithm is a spatiotemporal interpolation method applied to fill in missing or incomplete data using a low‐rank tensor structure. The results show that the accuracy (R2 = 0.880, RMSE = 0.037) of the proposed method is much higher than that of three traditional disease mapping methods: Cokriging, hierarchical Bayesian, and sandwich estimation methods. This study provides a new disease mapping approach to improve the quality and completeness of data in syndrome surveillance or other familiar systems with a large proportion of missing data.

Reconstruction and denoising of high-dimensional seismic data via Frobenius-nuclear mixed norm constraints

Abstract The seismic data acquisition design with ‘two-wide and one-high’ geometry effectively improves the imaging quality of seismic records. However, when data are acquired in the field, complex near surface conditions and environmental factors can introduce a variety of noises and gaps in seismic data, impacting the accuracy of seismic imaging. Currently, the method of low-rank matrix/tensor completion is commonly employed for data reconstruction after normal moveout (NMO). In a complex subsurface medium, common midpoint data processed with NMO may not satisfy the linear or quasi-linear assumptions within local data windows. Therefore, this paper exploits the inherent low-rank structure of high-dimensional data to propose a high-dimensional tensor completion method under the Frobenius-nuclear mixed norm constraint (FN-TC). This method unfolds the 4D data tensor into the frequency-space domain along its modes (m, n) and subsequently imposes a non-convex Frobenius-nuclear mixed norm constraint on the unfolded approximate matrices. This approach closely approximates the rank function of the factor matrices, thereby enhancing the accuracy of data modeling. Theoretical and practical studies demonstrate that the novel FN-TC approach can effectively reconstruct high-dimensional seismic data and suppress noise, thereby providing data support for subsequent high-precision seismic imaging.

A nonlinear high-order transformations-based method for high-order tensor completion

The high-order tensor nuclear norm model (HTNN) has recently shown promising results in tensor completion problems. The HTNN-based approaches rely on the low-rank structure of tensor slices during reversible transformations. However, under reversible transformations, the low-rank configuration within the slice-wise modality of the tensor’s structure is not markedly pronounced. In order to more effectively describe the low-rank characteristics, we propose a model based on the nonlinear high-order transform-based tensor nuclear norm (NHTNN). Specifically, our framework consists of a linearly semi-orthogonal transformation along the high-dimensional modality and an element-wise nonlinear transformation. We introduce a model for tensor completion, grounded in the suggested measure of tensor low-rank, i.e., NHTNN. Utilizing this non-convex nonlinear model, we formulate a proximal alternating minimization (PAM) algorithm, establishing its convergence through a rigorous proof. In experiments on datasets such as hyperspectral videos (HSVs) and color videos (CVs), our approach demonstrates superior quantitative numerical results and qualitative visual effects compared to cutting-edge tensor completion techniques.