Low-rank Matrix Completion Research Articles

Efficient low-rank quaternion matrix completion under the learnable transforms for color image recovery

Recent low-rank quaternion matrix completion (LRQMC) approaches have been extensively studied to recover missing data of color images. However, these methods need to frequently compute the quaternion singular value decompositions (QSVD) of the quaternion matrix, making them unsuitable for large-scale data. In this paper, we suggest an efficient LRQMC model based on the learnable transforms for color image recovery. The key idea is to project the large-scale quaternion matrix to a small-scale quaternion matrix via the semi-orthogonal transforms along each mode, which significantly reduces the computational cost of QSVD. We then apply a nonconvex approximation of rank (i.e., weighted Schatten p-norm) onto the small-scale quaternion matrix to achieve a better quaternion rank estimation. The alternating direction method of multipliers scheme is developed to solve the proposed model, and the weak convergence property of the algorithm is discussed. Experimental results on color images demonstrate that our method is considerably faster than state-of-art approaches while achieving comparative recovery performance.

Truncated quadratic norm minimization for bilinear factorization based matrix completion

Low-rank matrix completion is an important research topic with a wide range of applications. One prevailing way for matrix recovery is based on rank minimization. Directly solving this problem is NP hard. Therefore, various rank surrogates are developed, like nuclear norm. However, nuclear norm regularization minimizes the sum of all the singular values, and hence the rank is not well approximated. We propose a new rank substitution named truncated quadratic norm that performs the corresponding truncated quadratic operation on the singular values. This function takes the square of the minor singular values and maps large singular values to one. In order to reduce computational complexity, the original target matrix is factorized into two small matrices on which the truncated quadratic norm is imposed. The resultant problem is then solved by alternating minimization. We also prove that the solution sequence is able to converge to a critical point. Experimental results on synthetic data and real-world images demonstrate the excellent performance of our method in terms of recovery accuracy.

MDAlmc: A Novel Low-rank Matrix Completion Model for MiRNADisease Association Prediction by Integrating Similarities among MiRNAs and Diseases.

The importance of microRNAs (miRNAs) has been emphasized by an increasing number of studies, and it is well-known that miRNA dysregulation is associated with a variety of complex diseases. Revealing the associations between miRNAs and diseases are essential to disease prevention, diagnosis, and treatment. However, traditional experimental methods in validating the roles of miRNAs in diseases could be very expensive, labor-intensive and time-consuming. Thus, there is a growing interest in predicting miRNA-disease associations by computational methods. Though many computational methods are in this category, their prediction accuracy needs further improvement for downstream experimental validation. In this study, we proposed a novel model to predict miRNA-disease associations by low-rank matrix completion (MDAlmc) integrating miRNA functional similarity, disease semantic similarity, and known miRNA-disease associations. In the 5-fold cross-validation, MDAlmc achieved an average AUROC of 0.8709 and AUPRC of 0.4172, better than those of previous models. Among the case studies of three important human diseases, the top 50 predicted miRNAs of 96% (breast tumors), 98% (lung tumors), and 90% (ovarian tumors) have been confirmed by previous literatures. And the unconfirmed miRNAs were also validated to be potential disease-associated miRNAs. MDAlmc is a valuable computational resource for miRNA-disease association prediction.

Low-Rank Matrix Completion Theory via Plücker Coordinates

Despite the popularity of low-rank matrix completion, the majority of its theory has been developed under the assumption of random observation patterns, whereas very little is known about the practically relevant case of non-random patterns. Specifically, a fundamental yet largely open question is to describe patterns that allow for unique or finitely many completions. This paper provides three such families of patterns for any rank and any matrix size. A key to achieving this is a novel formulation of low-rank matrix completion in terms of Plüucker coordinates, the latter a traditional tool in computer vision. This connection is of potential significance to a wide family of matrix and subspace learning problems with incomplete data.

Adaptive thresholding for iterative matrix completion with heterogeneous missing probability: H-AdaptiveImpute

Several low-rank matrix completion algorithms have been developed, which often employ thresholded singular value decomposition (SVD) iteratively. Their performance depends on the choice of thresholding tuning parameters and initial input matrix. In particular, the initialization of the non-convex optimization algorithms plays a pivotal role. However, they often use a zero matrix or a consistent low-rank matrix estimator under the simple homogeneous observation probability assumption as the initial input matrix. This paper proposes a non-convex low-rank matrix completion algorithm with an intelligent initial matrix under the heterogeneous observation probability structure. Specifically, the proposed algorithm iterates the thresholded SVD starting from a tailored initialization based on the estimator of the singular values and vectors of the underlying matrix. We show that the proposed initial estimator is consistent, and the adaptive thresholding tuning parameters are estimated using theoretically justified and data-driven procedures. The results of the numerical experiments show that the proposed algorithm performs well compared to the existing method.

Low-Rank Matrix Sensing-Based Channel Estimation for mmWave and THz Hybrid MIMO Systems

This paper studies the channel estimation for wideband multiple-input multiple-output (MIMO) systems equipped with hybrid analog/digital transceivers operating in the millimeter-wave (mmWave) or terahertz (THz) bands. By exploiting the low-rank property of the concatenated channel matrix of the delay taps, we formulate the channel estimation problem as a low-rank matrix sensing (LRMS) problem and solve it using a low-complexity generalized conditional gradient-alternating minimization (GCG-ALTMIN) algorithm. This LRMS-based solution can accommodate different precoder/combiner and training structures. In addition, it does not require knowledge about the array responses at the transceivers, in contrast to most existing solutions allowing low training overhead. Furthermore, a preconditioned conjugate gradient (PCG) algorithm-based implementation and a low-rank matrix completion (LRMC) formulation are proposed to further reduce the computational complexity. In order to enhance the channel estimation performance for fat and tall channel matrices, we introduce a matrix reshaping approach that can preserve the channel rank by exploiting the shift-invariance property of uniform arrays. We also introduce a spectrum denoising (SD) approach for further improving the performance when the array responses are known and the number of paths is small. These approaches can effectively enhance the performance at a given training overhead. Simulation results suggest that the proposed solutions can achieve higher channel estimation accuracy and reduce the computational complexity as compared to several representative channel estimation schemes.

Matrix completion via modified schatten 2/3-norm

Low-rank matrix completion is a hot topic in the field of machine learning. It is widely used in image processing, recommendation systems and subspace clustering. However, the traditional method uses the nuclear norm to approximate the rank function, which leads to only the suboptimal solution. Inspired by the closed-form formulation of L_{2/3} regularization, we propose a new truncated schatten 2/3-norm to approximate the rank function. Our proposed regularizer takes full account of the prior rank information and achieves a more accurate approximation of the rank function. Based on this regularizer, we propose a new low-rank matrix completion model. Meanwhile, a fast and efficient algorithm are designed to solve the proposed model. In addition, a rigorous mathematical analysis of the convergence of the proposed algorithm is provided. Finally, the superiority of our proposed model and method is investigated on synthetic data and recommender system datasets. All results show that our proposed algorithm is able to achieve comparable recovery performance while being faster and more efficient than state-of-the-art methods.

Nonnegative Low Rank Matrix Completion by Riemannian Optimalization Methods

Nonnegative Low Rank Matrix Completion by Riemannian Optimalization Methods

Bayesian Uncertainty Quantification for Low-Rank Matrix Completion

We consider the problem of uncertainty quantification for an unknown low-rank matrix X, given a partial and noisy observation of its entries. This quantification of uncertainty is essential for many real-world problems, including image processing, satellite imaging, and seismology, providing a principled framework for validating scientific conclusions and guiding decision-making. However, existing literature has mainly focused on the completion (i.e., point estimation) of the matrix X, with little work on investigating its uncertainty. To this end, we propose in this work a new Bayesian modeling framework, called BayeSMG, which parametrizes the unknown X via its underlying row and column subspaces. This Bayesian subspace parametrization enables efficient posterior inference on matrix subspaces, which represents interpretable phenomena in many applications. This can then be leveraged for improved matrix recovery. We demonstrate the effectiveness of BayeSMG over existing Bayesian matrix recovery methods in numerical experiments, image inpainting, and a seismic sensor network application.

Repairing Artifacts in Neural Activity Recordings Using Low-Rank Matrix Estimation.

Electrophysiology recordings are frequently affected by artifacts (e.g., subject motion or eye movements), which reduces the number of available trials and affects the statistical power. When artifacts are unavoidable and data are scarce, signal reconstruction algorithms that allow for the retention of sufficient trials become crucial. Here, we present one such algorithm that makes use of large spatiotemporal correlations in neural signals and solves the low-rank matrix completion problem, to fix artifactual entries. The method uses a gradient descent algorithm in lower dimensions to learn the missing entries and provide faithful reconstruction of signals. We carried out numerical simulations to benchmark the method and estimate optimal hyperparameters for actual EEG data. The fidelity of reconstruction was assessed by detecting event-related potentials (ERP) from a highly artifacted EEG time series from human infants. The proposed method significantly improved the standardized error of the mean in ERP group analysis and a between-trial variability analysis compared to a state-of-the-art interpolation technique. This improvement increased the statistical power and revealed significant effects that would have been deemed insignificant without reconstruction. The method can be applied to any time-continuous neural signal where artifacts are sparse and spread out across epochs and channels, increasing data retention and statistical power.

Non-uniformly sampled 2D NMR Spectroscopy reconstruction based on Low Rank Hankel Matrix Fast Tri-Factorization and Non-convex Factorization

Nuclear magnetic resonance (NMR) spectroscopy is widely used in chemistry and medicine to research the composition of matter and the spatial structure of proteins. However, NMR signals with larger sizes and higher dimensions are slower to acquire. Therefore, Non-Uniform Sampling (NUS) methods are used to speed up the acquisition of NMR signals, and some mathematical algorithms are used to recover the original NMR signals from the NUS data. In recent years, low-rank Hankel matrix completion is proved to recover NMR data with less error. However, it requires singular value decomposition (SVD) with high computational complexity in each iteration, which is not suitable for reconstructing signals with large sizes. In this paper, we propose a Fast Tri-Factorization (FTF) method to decompose the low-rank Hankel matrix into three small-scale matrices, and it convert the original problem into a matrix nuclear norm minimization problem on the small-scale matrix to mitigate high computation cost of multiple SVDs. Furthermore, the factor matrix size remains constant for different size of the Hankel matrices. We further replace the low-rank constraint with Non-convex Factorization (NF) to avoid the time-consuming SVD in each iteration. Numerical experiments show that the proposed method can significantly speed up the NMR reconstruction compared with several existing algorithms while ensuring the reconstruction accuracy.

Quaternion Matrix Factorization for Low-Rank Quaternion Matrix Completion

The main aim of this paper is to study quaternion matrix factorization for low-rank quaternion matrix completion and its applications in color image processing. For the real-world color images, we proposed a novel model called low-rank quaternion matrix completion (LRQC), which adds total variation and Tikhonov regularization to the factor quaternion matrices to preserve the spatial/temporal smoothness. Moreover, a proximal alternating minimization (PAM) algorithm was proposed to tackle the corresponding optimal problem. Numerical results on color images indicate the advantages of our method.

A low-rank deep image prior reconstruction for free-breathing ungated spiral functional CMR at 0.55T and 1.5T.

This study combines a deep image prior with low-rank subspace modeling to enable real-time (free-breathing and ungated) functional cardiac imaging on a commercial 0.55T scanner. The proposed low-rank deep image prior (LR-DIP) uses two u-nets to generate spatial and temporal basis functions that are combined to yield dynamic images, with no need for additional training data. Simulations and scans in 13 healthy subjects were performed at 0.55T and 1.5T using a golden angle spiral bSSFP sequence with images reconstructed using [Formula: see text]-ESPIRiT, low-rank plus sparse (L + S) matrix completion, and LR-DIP. Cartesian breath-held ECG-gated cine images were acquired for reference at 1.5T. Two cardiothoracic radiologists rated images on a 1-5 scale for various categories, and LV function measurements were compared. LR-DIP yielded the lowest errors in simulations, especially at high acceleration factors (R [Formula: see text] 8). LR-DIP ejection fraction measurements agreed with 1.5T reference values (mean bias -0.3% at 0.55T and -0.2% at 1.5T). Compared to reference images, LR-DIP images received similar ratings at 1.5T (all categories above 3.9) and slightly lower at 0.55T (above 3.4). Feasibility of real-time functional cardiac imaging using a low-rank deep image prior reconstruction was demonstrated in healthy subjects on a commercial 0.55T scanner.

An Improved Low-Rank Matrix Fitting Method Based on Weighted L1,p Norm Minimization for Matrix Completion

Low-rank matrix completion, which aims to recover a matrix with many missing values, has attracted much attention in many fields of computer science. A low-rank matrix fitting (LMaFit) method has been proposed for fast matrix completion recently. However, this method cannot converge accurately on matrices of real-world images. For improving the accuracy of LMaFit method, an improved low-rank matrix fitting (ILMF) method based on the weighted [Formula: see text] norm minimization is proposed in this paper, where the [Formula: see text] norm is the summation of the [Formula: see text]-power [Formula: see text] of [Formula: see text] norms of rows in a matrix. In the proposed method, i.e. the ILMF method, the incomplete matrix that may be corrupted by noises is decomposed into the summation of a low-rank matrix and a noise matrix at first. Then, a weighted [Formula: see text] norm minimization problem is solved by using an alternating direction method for improving the accuracy of matrix completion. Experimental results on real-world images show that the ILMF method has much better performances in terms of both the convergence accuracy and convergence speed than the compared methods.

Weighted hybrid truncated norm regularization method for low-rank matrix completion

Weighted hybrid truncated norm regularization method for low-rank matrix completion

Channel Estimation for RIS-Aided mmWave Massive MIMO System Using Few-Bit ADCs

Millimeter wave (mmWave) massive multiple-input multiple-output (massive MIMO) is one of the most promising technologies for the fifth generation and beyond wireless communication system. However, a large number of antennas incur high power consumption and hardware costs, and high-frequency communications place a heavy burden on the analog-to-digital converters (ADCs) at the base station (BS). Furthermore, it is too costly to equipping each antenna with a high-precision ADC in a large antenna array system. It is promising to adopt low-resolution ADCs to address this problem. In this letter, we investigate the cascaded channel estimation for a mmWave massive MIMO system aided by a reconfigurable intelligent surface (RIS) with the BS equipped with few-bit ADCs. Due to the low-rank property of the cascaded channel, the estimation of the cascaded channel can be formulated as a low-rank matrix completion problem. We introduce the Bayesian optimal inference framework to tackle with the information loss caused by quantization. To implement the estimator and achieve the matrix completion, we use efficient bilinear generalized approximate message passing (BiG-AMP) algorithm. Extensive simulation results verify that our proposed method can accurately estimate the cascaded channel for the RIS-aided mmWave massive MIMO system with low-resolution ADCs.

Low-Rank Matrix Completion via QR-Based Retraction on Manifolds

Low-rank matrix completion aims to recover an unknown matrix from a subset of observed entries. In this paper, we solve the problem via optimization of the matrix manifold. Specially, we apply QR factorization to retraction during optimization. We devise two fast algorithms based on steepest gradient descent and conjugate gradient descent, and demonstrate their superiority over the promising baseline with the ratio of at least 24%.

Simulation comparisons between Bayesian and de-biased estimators in low-rank matrix completion

In this paper, we study the low-rank matrix completion problem, a class of machine learning problems, that aims at the prediction of missing entries in a partially observed matrix. Such problems appear in several challenging applications such as collaborative filtering, image processing, and genotype imputation. We compare the Bayesian approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optimal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally interesting results that the de-biased estimator is just as good as the Bayesian estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. In addition, we also find that the empirical coverage rate of the confidence intervals obtained by the de-biased estimator for an entry is absolutely lower than of the considered credible interval. These results suggest further theoretical studies on the estimation error and the concentration of Bayesian methods as they are quite limited up to present.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

Robust Matrix Completion for Elliptic Positioning in the Presence of Outliers and Missing Data

Elliptic target positioning from the bistatic ranges (BRs), as an emerging localization scheme, has recently gained considerable traction for its diverse applications in multistatic systems such as radar, sonar, and wireless sensor networks. This contribution extends the work of previous research on the low-rank property of the BR matrix (Xiong, “Denoising of bistatic ranges for elliptic positioning,” IEEE Geosci. Remote Sens. Lett., vol. 20, pp. 1–3, 2023, Art. no. 3500503) to the brand new use case of robust elliptic positioning in the presence of missing data. Due to the structures of the outlier-inducing errors when embodied in the BR matrix, many of the off-the-shelf low-rank matrix completion (LRMC) solutions cannot be applied. We address this challenge by formulating the problem of outlier-resistant BR matrix recovery as constrained minimization of an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> -norm based loss function, and devising an algorithm based on alternating direction method of multipliers to efficiently solve the resultant LRMC. Simulations are conducted to demonstrate the efficacy of the developed robust elliptic positioning technique in various localization scenarios.