Low-rank Matrix Completion Research Articles

Mode Measurement of Vortex Electromagnetic Wave With Random Receiving Array Element Defects via Low Rank Matrix Completion

Mode Measurement of Vortex Electromagnetic Wave With Random Receiving Array Element Defects via Low Rank Matrix Completion

[formula omitted]-norm regularized quaternion matrix completion using sparse representation and approximate QSVD

Color image completion is a challenging problem in computer vision. Despite the effectiveness of low-rank quaternion matrix completion (LRQMC) methods, their reliance on quaternion singular value decompositions (QSVD) makes them computationally expensive. We propose a novel method based on quaternion Qatar Riyal decomposition (QQR) and quaternion L2,1-norm, named QLNM-QQR, which reduces computational complexity by avoiding the need for calculating the QSVD of large quaternion matrices. Furthermore, we introduce two improvements: IRQLNM-QQR, which utilizes iteratively reweighted quaternion L2,1-norm minimization, and QLNM-QQR-SR, which integrates sparse regularization. Additionally, an analysis of the convergence of the proposed algorithms is conducted. Our experiments show that IRQLNM-QQR outperforms QLNM-QQR on both natural and medical color images. The QLNM-QQR-SR method achieves higher PSNR and SSIM values than state-of-the-art methods, with PSNR averaging 4-8 dB higher than non-sparse methods and 0.4 dB higher than the sparse-based TNN-SR method.

Two-Dimensional Direction Finding for L-Shaped Coprime Array via Minimization of the Ratio of the Nuclear Norm and the Frobenius Norm

More recently, the ability of the coprime array to yield large array apertures and high degrees of freedom in comparison with the uniform linear array (ULA) has drawn an enormous amount of attention. In light of this, we propose a low-rank matrix completion algorithm via minimization of the ratio of the nuclear norm and the Frobenius norm (N/F) to solve the two-dimensional (2D) direction finding problem for the L-shaped coprime array (LsCA). Specifically, we first interpolate the virtual co-array signal related to the cross-correlation matrix (CCM) and utilize the interpolated virtual signal for Toeplitz matrix reconstruction. Then, the N/F method is employed to perform low-rank matrix completion on the reconstructed matrix. Finally, exploiting the conjugate symmetry characteristics of the completed matrix, we further develop a direction-finding algorithm that enables 2D angle estimation. Remarkably, the 2D angles are able to be automatically paired by the proposed algorithm. Numerical simulation findings demonstrate that the proposed N/F algorithm generates excellent angular resolution and computational complexity. Furthermore, this algorithm yields better estimation accuracy compared to the competing algorithms.

Robust Matrix Completion with Heavy-Tailed Noise

This article studies noisy low-rank matrix completion in the presence of heavy-tailed and possibly asymmetric noise, where we aim to estimate an underlying low-rank matrix given a set of highly incomplete noisy entries. Though the matrix completion problem has attracted much attention in the past decade, there is still lack of theoretical understanding when the observations are contaminated by heavy-tailed noises. Prior theory falls short of explaining the empirical results and is unable to capture the optimal dependence of the estimation error on the noise level. In this article, we adopt an adaptive Huber loss to accommodate heavy-tailed noise, which is robust against large and possibly asymmetric errors when the parameter in the Huber loss function is carefully designed to balance the Huberization biases and robustness to outliers. Then, we propose an efficient nonconvex algorithm via a balanced low-rank Burer-Monteiro matrix factorization and gradient descent with robust spectral initialization. We prove that under merely a bounded second-moment condition on the error distributions, rather than the sub-Gaussian assumption, the Euclidean errors of the iterates generated by the proposed algorithm decrease geometrically fast until achieving a minimax-optimal statistical estimation error, which has the same order as that in the sub-Gaussian case. The key technique behind this significant advancement is a powerful leave-one-out analysis framework. The theoretical results are corroborated by our numerical studies. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Robust low-rank matrix completion via sparsity-inducing regularizer

This paper proposes a sparsity-inducing regularizer associated with the Welsch function. We theoretically show that the regularizer is quasiconvex and the corresponding Moreau envelope is convex. Moreover, the closed-form solution to its Moreau envelope, namely, the proximity operator, is derived. Unlike conventional nonconvex regularizers like the ℓp-norm with 0<p<1 that generally needs iterations to obtain the corresponding proximity operator, the developed regularizer has a closed-form proximity operator. We utilize our regularizer to penalize the singular values as well as sparse outliers of the distorted data, and develop an efficient algorithm for robust matrix completion. Convergence of the suggested method is analyzed and we prove that any accumulation point is a stationary point. Finally, experimental results demonstrate that our algorithm is superior to the competing techniques in terms of restoration performance. MATALB codes are available at https://github.com/bestzywang/RMC-NNSR.

Efficient quantum state estimation with low-rank matrix completion

This paper introduces a novel and efficient technique for quantum state estimation, coined as low-rank matrix-completion quantum state tomography for characterizing pure quantum states, as it requires only non-entangling bases and 2n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2n + 1$\\end{document} local Pauli operators. This significantly reduces the complexity of the process and increases the accuracy of the state estimation, as it eliminates the need for the entangling bases, which are experimentally difficult to implement on quantum devices. The required minimal post-processing, improved accuracy and efficacy of this matrix-completion-based method make it an ideal benchmarking tool for investigating the properties of quantum systems, enabling researchers to verify the accuracy of quantum devices, characterize their performance, and explore the underlying physics of quantum phenomena. Our numerical results demonstrate that this method outperforms contemporary techniques in its ability to accurately reconstruct multi-qubit quantum states on real quantum devices, making it an invaluable contribution to the field of quantum state characterization and an essential step toward the reliable deployment of intermediate- and large-scale quantum devices.

Distributed credible evidence fusion with privacy-preserving

Considering data safety in more and more applied peer-to-peer networks, such as wireless sensor networks, has become the focus of information fusion, this paper proposes the problem of credible evidence fusion (CEF) in a distributed system with privacy-preserving, where agent’s raw evidence is shared only with authenticated neighbors while access or inference by non-neighbors is prevented. This privacy-preserving in a distributed style brings out two challenges never met in centralized CEF. One is about credibility calculation consensus (CCC), where the local evidence difference measure matrix (EDMM) faces elements missing for evidence unavailability to non-neighbors. The other is about privacy-preserving fusion consensus (PFC), because fusion based on raw evidence may lead to a counter-intuitive result. In CCC, missing elements in EDMM are recovered via low-rank matrix completion, and credibility consistency is further guaranteed via diffusion gradient descent. In PFC, a new measure named privacy iterative compensation term (PICT) is constructed based on credibility and discounted evidence so that raw evidence is impossibly inferred. Furthermore, the linear average consensus based on PICT is presented and proved to be converging to the centralized CEF if credibility is accurately given. The computational complexity is proportional to the cube of the number of agents and the square of the number of elements in the power set of the framework of discrimination. Simulation results demonstrate that the proposed method achieves credibility errors below 0.1, leading to fusion results and classification accuracy approaching those of centralized CEF, with a 30% improvement compared to RANSAC-based and COF-based methods.

A Novel Nonconvex Relaxation Approach to Low-Rank Matrix Completion of Inexact Observed Data

A Novel Nonconvex Relaxation Approach to Low-Rank Matrix Completion of Inexact Observed Data

Low-rank reconstruction for simultaneous double half-echo 23Na and undersampled 23Na multi-quantum coherences MRI.

To develop a new sequence to simultaneously acquire Cartesian sodium (23Na) MRI and accelerated Cartesian single (SQ) and triple quantum (TQ) sodium MRI of in vivo human brain at 7 T by leveraging two dedicated low-rank reconstruction frameworks. The Double Half-Echo technique enables short echo time Cartesian 23Na MRI and acquires two k-space halves, reconstructed by a low-rank coupling constraint. Additionally, three-dimensional (3D) 23Na Multi-Quantum Coherences (MQC) MRI requires multi-echo sampling paired with phase-cycling, exhibiting a redundant multidimensional space. Simultaneous Autocalibrating and k-Space Estimation (SAKE) were used to reconstruct highly undersampled 23Na MQC MRI. Reconstruction performance was assessed against five-dimensional (5D) CS, evaluating structural similarity index (SSIM), root mean squared error (RMSE), signal-to-noise ratio (SNR), and quantification of tissue sodium concentration and TQ/SQ ratio in silico, in vitro, and in vivo. The proposed sequence enabled the simultaneous acquisition of fully sampled 23Na MRI while leveraging prospective undersampling for 23Na MQC MRI. SAKE improved TQ image reconstruction regarding SSIM by 6% and reduced RMSE by 35% compared to 5D CS in vivo. Thanks to prospective undersampling, the spatial resolution of 23Na MQC MRI was enhanced from mm3 to mm3 while reducing acquisition time from min to min. The proposed sequence, coupled with low-rank reconstructions, provides an efficient framework for comprehensive whole-brain sodium MRI, combining TSC, T2*, and TQ/SQ ratio estimations. Additionally, low-rank matrix completion enables the reconstruction of highly undersampled 23Na MQC MRI, allowing for accelerated acquisition or enhanced spatial resolution.

Sparse graph cascade multi-kernel fusion contrastive learning for microbe–disease association prediction

Predicting microbe–disease associations (MDA) is crucial for proactively demystifying diseases causes and preventing them. Traditional prediction methods endure labor-intensive, time-consuming, and expensive. Therefore, this paper proposes CasMF-GCL, a novel Graph Contrastive Learning model based on sparse relation augmentation and a Cascaded Multi-kernel Fusion Network for MDA prediction. CasMF-GCL maximizes the use of sparse correlation information and eliminates data noise through an association augmentation technique based on low-rank sparse matrix completion. We employ diverse strategies to enhance the features of heterogeneous networks by exploring the similarity between microbes and diseases. Our model employs a new multi-layer graph convolutional network variant with a cascaded multi-kernel fusion mechanism, enabling weighted coding of local and global features for diseases and microbes. To further improve performance, we incorporate a self-supervised contrastive learning schema using multi-grain disease and microbe features and large-scale relation augmentations. Extensive experiments on two renowned datasets demonstrate that CasMF-GCL outperforms current state-of-the-art methods in seven indexes. The AUC values from 5-fold cross-validation are 0.997 and 0.989 for the two datasets, respectively. Ablation studies confirm the effectiveness of graph data augmentation, the power of the contrastive strategy, and the indispensability of the cascaded multi-kernel fusion network. Furthermore, case studies validate the prediction performance of CasMF-GCL.

Robust Tensor-Based DOA and Polarization Estimation in Conformal Polarization Sensitive Array with Bad Data.

Partially impaired sensor arrays pose a significant challenge in accurately estimating signal parameters. The occurrence of bad data is highly probable, resulting in random loss of source information and substantial performance degradation in parameter estimation. In this paper, a tensor variational sparse Bayesian learning (TVSBL) method is proposed for the estimate of direction of arrival (DOA) and polarization parameters jointly based on a conformal polarization sensitive array (CPSA), taking into account scenarios with the partially impaired sensor array. First, a sparse tensor-based received data model is developed for CPSAs that incorporates bad data. Then, a column vector detection method is proposed to diagnose the positions of the impaired sensors. In scenarios involving partially impaired sensor arrays, a low-rank matrix completion method is employed to recover the random loss of signal information. Finally, variational sparse Bayesian learning (VSBL) and minimum eigenvector methods are utilized sequentially to obtain the DOA and polarization parameters estimation, successively. Furthermore, the Cramér-Rao bound is given for the proposed method. Simulation results validated the effectiveness of the proposed method.

Orthogonal subspace exploration for matrix completion

Matrix completion, aiming at restoring a low-rank matrix from observed entries, indicates the connection with the subspace clustering due to the low-rank property. However, it is expensive to incorporate subspace learning into the pervasive surrogate of matrix completion, the nuclear norm. In this paper, we design an orthogonal subspace exploration model for matrix completion, which can be easily integrated due to the succinct formulation. Then, we propose a non-convex surrogate with tractable solutions for low-rank matrix completion, so that the subspace exploration can be performed simultaneously. Compared with the existing surrogates (e.g., nuclear norm, Schatten-p norm, max norm, etc.), the proposed surrogate is differential such that the optimization is still simple even after the subspace exploration is incorporated. Although the surrogate is non-convex, a parameter-free algorithm that is proved to converge into the global optimum is developed. The optimization consists of closed-form solutions so that the orthogonal subspace exploration will not distinctly bring additional costs and the algorithm empirically converges within dozens of iterations. Experiments illustrate the efficiency and superiority of our model.

A Designed Thresholding Operator for Low-Rank Matrix Completion

In this paper, a new thresholding operator, namely, designed thresholding operator, is designed to recover the low-rank matrices. With the change of parameter in designed thresholding operator, the designed thresholding operator can apply less bias to the larger singular values of a matrix compared with the classical soft thresholding operator. Based on the designed thresholding operator, an iterative thresholding algorithm for recovering the low-rank matrices is proposed. Numerical experiments on some image inpainting problems show that the proposed thresholding algorithm performs effectively in recovering the low-rank matrices.

Ocean sound speed field reconstruction with graph regularized deep matrix factorization

To reconstruct the ocean sound speed field (SSF), a matrix in horizontal could be formed by gridding the sea area, where some sparse entries were filled by the observation sound speed profiles (SSPs). The reconstruction of the SSF can be modeled as a matrix completion problem, using limited and noisy observation values to reconstruct the entire matrix. Furthermore, the spatial correlation of the SSF could be modeled by graph space model, utilizing with the low-rank property of the SSF matrix, the problem of graph regularization low-rank matrix completion was developed. The observed average SSPs in six cross section were estimated from the four underwater acoustic tomography moorings, and nine local SSPs were obtained from pressure inverted echo sounders (PIESs) inversion. By combining the method of deep matrix factorization and the graph regularization, the Graph Regularized Deep Matrix Factorization (GRDMF) method was proposed. The simulation results showed that the GRDMF performs better than the spatial interpolation algorithm in SSF reconstruction. Considering the application scenarios of deep-sea acoustic tomography, an average constraint (AC) item was designed additionally, and then a customized algorithm named GRDMF-AC is proposed. Finally, data of the experiment at the South China Sea in 2021 is processed to verify GRDMF-AC, the results showed that the reconstructed SSF captures mesoscale eddy in the experiment.

Sensing and Deep CNN-Assisted Semi-Blind Detection for Multi-User Massive MIMO Communications

Attaining precise target detection and channel measurements are critical for guiding beamforming optimization and data demodulation in massive multiple-input multiple-output (MIMO) communication systems with hybrid structures, which requires large pilot overhead as well as substantial computational complexity. With benefits from the powerful detection characteristics of MIMO radar, we aim for designing a novel sensing-assisted semi-blind detection scheme in this paper, where both the inherent low-rankness of signal matrix and the essential knowledge about geometric environments are fully exploited under a designated cooperative manner. Specifically, to efficiently recover the channel factorizations via the formulated low-rank matrix completion problem, a low-complexity iterative algorithm stemming from the alternating steepest descent (ASD) method is adopted to obtain the solutions in case of unknown noise statistics. Moreover, we take one step forward by employing the denoising convolutional neural network (DnCNN) to preprocess the received signals due to its favorable performance of handling Gaussian denoising. The overall paradigm of our proposed scheme consists of three stages, namely (1) target parameter sensing, (2) communication signal denoising and (3) semi-blind detection refinement. Simulation results show that significant estimation gains can be achieved by the proposed scheme with reduced training overhead in a variety of system settings.

Fermionic Reduced Density Low-Rank Matrix Completion, Noise Filtering, and Measurement Reduction in Quantum Simulations.

Fermionic reduced density matrices summarize the key observables in Fermionic systems. In electronic systems, the two-particle reduced density matrix (2-RDM) is sufficient to determine the energy and most physical observables of interest. Here, we consider the possibility of using matrix completion to reconstruct the two-particle reduced density matrix to chemical accuracy from partial information. We consider the case of noiseless matrix completion, where the partial information corresponds to a subset of the 2-RDM elements, as well as noisy completion, where the partial information corresponds to both a subset of elements and statistical noise in their values. Through experiments on a set of 24 molecular systems, we find that 2-RDM can be efficiently reconstructed from a reduced amount of information. In the case of noisy completion, this results in a multiple orders of magnitude reduction in the number of measurements needed to determine the 2-RDM with chemical accuracy. These techniques can be readily applied to both classical and quantum algorithms for quantum simulations.

Maximal matroids in weak order posets

Let X be a family of subsets of a finite set E. A matroid on E is called an X-matroid if each set in X is a circuit. We develop techniques for determining when there exists a unique maximal X-matroid in the weak order poset of all X-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families X and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.

NT-DPTC: A non-negative temporal dimension preserved tensor completion model for missing traffic data imputation

Missing traffic data imputation is an important step in the intelligent transportation systems. Low rank approximation is an important method for the missing traffic data imputation, especially the low rank matrix/tensor completion. However, the low-rank matrix completion has a huge time cost and the low-rank tensor completion cannot fully extract the information and maintain non-negativity. Therefore, we propose a novel non-negative temporal dimension preserved tensor completion (NT-DPTC) model with ideas: a) proposing a novel dimension preserved (DP) tensor decomposition method, which decomposes a tensor into three latent factor tensors to fully extract the intrinsic feature, b) using a Sigmoid mapper for releasing non-negative constraint from the training process and increasing flexibility, c) exploiting temporal constrains and AdamW schemes for obtaining a higher accuracy and faster convergence. Experiments on four industrial application scenarios highlight the superiority of our proposed model when compared with the existing state-of-the-art models.

Harmonic retrieval using weighted lifted-structure low-rank matrix completion

In this paper, we investigate the problem of recovering the frequency components of a mixture of K complex sinusoids from a random subset of N equally-spaced time-domain samples. Because of the random subset, the samples are effectively non-uniform. Besides, the frequency values of each of the K complex sinusoids are assumed to vary continuously within a given range. For this problem, we propose a two-step strategy: (i) we first lift the incomplete set of uniform samples (unavailable samples are treated as missing data) into a structured matrix with missing entries, which is potentially low-rank; then (ii) we complete the matrix using a weighted nuclear minimization problem. We call the method a weighted lifted-structured (WLi) low-rank matrix recovery. Our approach can be applied to a range of matrix structures such as Hankel and double-Hankel, among others, and provides improvement over the unweighted existing schemes such as EMaC and DEMaC. We provide theoretical guarantees for the proposed method, as well as numerical simulations in both noiseless and noisy settings. Both the theoretical and the numerical results confirm the superiority of the proposed approach.

Low-Rank Matrix Completion with Poisson Observations via Nuclear Norm and Total Variation Constraints

Low-Rank Matrix Completion with Poisson Observations via Nuclear Norm and Total Variation Constraints