Low-rank Matrix Completion Problem Research Articles

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.

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.

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.

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

MNNMDA: Predicting human microbe-disease association via a method to minimize matrix nuclear norm

Identifying the potential associations between microbes and diseases is the first step for revealing the pathological mechanisms of microbe-associated diseases. However, traditional culture-based microbial experiments are expensive and time-consuming. Thus, it is critical to prioritize disease-associated microbes by computational methods for further experimental validation. In this study, we proposed a novel method called MNNMDA, to predict microbe-disease associations (MDAs) by applying a Matrix Nuclear Norm method into known microbe and disease data. Specifically, we first calculated Gaussian interaction profile kernel similarity and functional similarity for diseases and microbes. Then we constructed a heterogeneous information network by combining the integrated disease similarity network, the integrated microbe similarity network and the known microbe-disease bipartite network. Finally, we formulated the microbe-disease association prediction problem as a low-rank matrix completion problem, which was solved by minimizing the nuclear norm of a matrix with a few regularization terms. We tested the performances of MNNMDA in three datasets including HMDAD, Disbiome, and Combined Data with small, medium and large sizes respectively. We also compared MNNMDA with 5 state-of-the-art methods including KATZHMDA, LRLSHMDA, NTSHMDA, GATMDA, and KGNMDA, respectively. MNNMDA achieved area under the ROC curves (AUROC) of 0.9536 and 0.9364 respectively on HDMAD and Disbiome, better than the AUCs of compared methods under the 5-fold cross-validation for all microbe-disease associations. It also obtained a relatively good performance with AUROC 0.8858 in the combined data. In addition, MNNMDA was also better than other methods in area under precision and recall curve (AUPR) under the 5-fold cross-validation for all associations, and in both AUROC and AUPR under the 5-fold cross-validation for diseases and the 5-fold cross-validation for microbes. Finally, the case studies on colon cancer and inflammatory bowel disease (IBD) also validated the effectiveness of MNNMDA. In conclusion, MNNMDA is an effective method in predicting microbe-disease associations. AvailabilityThe codes and data for this paper are freely available at Github https://github.com/Haiyan-Liu666/MNNMDA.

Computational Prediction of Drug-Disease Association Based on Graph-Regularized One Bit Matrix Completion.

Investigation of existing drugs is an effective alternative to the discovery of new drugs for treating diseases. This task of drug re-positioning can be assisted by various kinds of computational methods to predict the best indication for a drug given the open-source biological datasets. Owing to the fact that similar drugs tend to have common pathways and disease indications, the association matrix is assumed to be of low-rank structure. Hence, the problem of drug-disease association prediction can be modeled as a low-rank matrix completion problem. In this work, we propose a novel matrix completion framework that makes use of the side-information associated with drugs/diseases for the prediction of drug-disease indications modeled as neighborhood graph: Graph regularized 1-bit matrix completion (GR1BMC). The algorithm is specially designed for binary data and uses parallel proximal algorithm to solve the aforesaid minimization problem taking into account all the constraints including the neighborhood graph incorporation and restricting predicted scores within the specified range. The results have been validated on two standard databases by evaluating the AUC across the 10-fold cross-validation splits. The usage of the method is also evaluated through a case study where top 5 indications are predicted for novel drugs, which then are verified with the CTD database.

Seismic data interpolation using nonlocal self-similarity prior

The use of a nonlocal self-similarity (NSS) prior, which refers to each reference patch always having many nonlocal similar patches, has demonstrated its effectiveness in seismic data random noise attenuation because of the repetitiveness of textures and structures in their global position. However, NSS-based approaches face challenges when dealing with seismic interpolation. In the presence of missing traces, similar patch matching may be highly unreliable, resulting in a limited performance of interpolation. To solve this problem, a two-stage iterative seismic-interpolation framework based on a rank-reduction (RR) algorithm is developed. In the first stage, preinterpolation seismic data are used to guide the similar patch matching, and the problem of missing trace recovery for the stacked matched patches is converted to the problem of low-rank matrix completion. In the second stage, the similar patches are directly searched on the interpolation result after stage 1 without external help; that is, exploiting its own NSS, which can achieve enhanced interpolation performance. For each iteration, we obtain accurate similarly matched groups and apply a simple and efficient truncated singular value decomposition for RR. Owing to the unique construction method of a low-rank matrix formed by similar patches, our approach can handle irregularly or regularly sampled seismic data. Numerical experiments verify the effectiveness of our method, compared with the curvelet, low-rank matrix fitting, and f- x prediction filtering methods.

Clustering for Topological Interference Management

To reduce the overhead and complexity of channel state information acquisition in interference alignment, the topological interference management (TIM) was proposed to manage interference, which only relied on the network topology information. The previous research on topological interference management via the low-rank matrix completion approach is known to be NP-hard. This paper considers the clustering method for the topological interference management problem, namely, the low-rank matrix completion for TIM is applied within each cluster. Based on the clustering result, we solve the low-rank matrix completion problem via nuclear norm minimization and Frobenius norm minimization function. Simulation results demonstrate that the proposed clustering method combined with TIM leads to significant gain on the achievable degrees of freedom.

Low Tucker rank tensor completion using a symmetric block coordinate descent method

AbstractLow Tucker rank tensor completion has wide applications in science and engineering. Many existing approaches dealt with the Tucker rank by unfolding matrix rank. However, unfolding a tensor to a matrix would destroy the data's original multi‐way structure, resulting in vital information loss and degraded performance. In this article, we establish a relationship between the Tucker ranks and the ranks of the factor matrices in Tucker decomposition. Then, we reformulate the low Tucker rank tensor completion problem as a multilinear low rank matrix completion problem. For the reformulated problem, a symmetric block coordinate descent method is customized. For each matrix rank minimization subproblem, the classical truncated nuclear norm minimization is adopted. Furthermore, temporal characteristics in image and video data are introduced to such a model, which benefits the performance of the method. Numerical simulations illustrate the efficiency of our proposed models and methods.

Topological Pilot Assignment in Large-Scale Distributed MIMO Networks

We consider the pilot assignment problem in large-scale distributed multi-input multi-output (MIMO) networks, where a large number of remote radio head (RRH) antennas are randomly distributed in a wide area, and jointly serve a relatively smaller number of users (UE) coherently. By artificially imposing structures on the UE-RRH connectivity, we model the network by a partially-connected interference network, so that the pilot assignment problem can be cast as a topological interference management problem with multiple groupcast messages. Building upon such connection, we formulate the topological pilot assignment (TPA) problem in two different ways with respect to whether or not the to-be-estimated channel connectivity pattern is known <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> . When it is known, we formulate the TPA problem as a low-rank matrix completion problem that can be solved by a simple alternating projection algorithm. Otherwise, we formulate it as a sequential maximum weight induced matching problem that can be solved by either a mixed integer linear program or a simple yet efficient greedy algorithm. With respect to two different formulations of the TPA problem, we evaluate the efficiency of the proposed algorithms under the cell-free massive MIMO setting.

Bayesian robust hankel matrix completion with uncertainty modeling for synchrophasor data recovery

Synchrophasor data suffer from quality issues like missing and bad data. Exploiting the low-rankness of the Hankel matrix of the synchrophasor data, this paper formulates the data recovery problem as a robust low-rank Hankel matrix completion problem and proposes a Bayesian data recovery method that estimates the posterior distribution of synchrophasor data from partial observations. In contrast to the deterministic approaches, our proposed Bayesian method provides an uncertainty index to evaluate the confidence of each estimation. To the best of our knowledge, this is the first method that provides confidence measure for synchrophasor data recovery. Numerical experiments on synthetic data and recorded synchrophasor data demonstrate that our method outperforms existing low-rank matrix completion methods.

A Nonconvex Method to Low-Rank Matrix Completion

In recent years, the problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion problem, has attracted much attention in many applications. However, it is a NP-hard problem due to the nonconvexity nature of the matrix rank function. In this paper, a rank Laplace function is studied to recover the low-rank matrices. Firstly, we propose an iterative Laplace thresholding algorithm to solve the regularized Laplace low-rank matrix completion problem. Secondly, some other iterative thresholding algorithms are designed to recover the low-rank matrices. Finally, we provide a series of numerical simulations to test the proposed algorithms on some low-rank matrix completion and image inpainting problems, and the results show that our algorithms perform better than some state-of-art methods in recovering the low-rank matrices.

A Riemannian rank-adaptive method for low-rank matrix completion

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.

Low-rank matrix completion in a general non-orthogonal basis

This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach. We introduce a condition on the basis called the correlation condition. This condition can be computed in time O(n3) and holds for many cases of deterministic basis. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter ν, under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from O(nrνlog2⁡n) uniformly random expansion coefficients.

Repositioning Drugs on Human Influenza A Viruses Based on a Novel Nuclear Norm Minimization Method.

Influenza A viruses, especially H3N2 and H1N1 subtypes, are viruses that often spread among humans and cause influenza pandemic. There have been several big influenza pandemics that have caused millions of human deaths in history, and the threat of influenza viruses to public health is still serious nowadays due to the frequent antigenic drift and antigenic shift events. However, only few effective anti-flu drugs have been developed to date. The high development cost, long research and development time, and drug side effects are the major bottlenecks, which could be relieved by drug repositioning. In this study, we proposed a novel antiviral Drug Repositioning method based on minimizing Matrix Nuclear Norm (DRMNN). Specifically, a virus-drug correlation database consisting of 34 viruses and 205 antiviral drugs was first curated from public databases and published literature. Together with drug similarity on chemical structure and virus sequence similarity, we formulated the drug repositioning problem as a low-rank matrix completion problem, which was solved by minimizing the nuclear norm of a matrix with a few regularization terms. DRMNN was compared with three recent association prediction algorithms. The AUC of DRMNN in the global fivefold cross-validation (fivefold CV) is 0.8661, and the AUC in the local leave-one-out cross-validation (LOOCV) is 0.6929. Experiments have shown that DRMNN is better than other algorithms in predicting which drugs are effective against influenza A virus. With H3N2 as an example, 10 drugs most likely to be effective against H3N2 viruses were listed, among which six drugs were reported, in other literature, to have some effect on the viruses. The protein docking experiments between the chemical structure of the prioritized drugs and viral hemagglutinin protein also provided evidence for the potential of the predicted drugs for the treatment of influenza.

Tensor Methods for Nonlinear Matrix Completion

In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. Th...

Two Novel Algorithms for Low-Rank Matrix Completion Problem

Low-rank matrix completion has been exploited in many signal processing problems over the past few years. It has been shown that nuclear norm minimization can retrieve such low-rank matrices via uniform sampling under coherence conditions. However, in many applications such as the Netflix problem, the low-rank matrix is not coherent, hence nuclear norm minimization is unable to solve the problem because of uniform sampling. To overcome this issue, we propose two algorithms to recover both coherent and incoherent low-rank matrices. In the first algorithm, it is assumed that we have a limited sampling budget that is associated with the available number of samples. Using a portion of the sampling budget, we propose an Uncertainty Information (UI) metric to quantify the uncertainty that we have for each element and consider the elements with the most uncertainty as the most informative elements to sample. Finally, the simulations results are performed to investigate the performance of the proposed algorithms and show their superiority to conventional approaches such as nuclear norm minimization Candes and Recht, 2009 and two-phase sampling algorithm Chen et al. 2015.

저 차원 Hankel 구조를 이용한 PMU 데이터 복구에 관한 연구

Phasor Measurement Units (PMUs) provide synchronized phasor measurements at much higher sampling rate than that in the traditional Supervisory Control And Data Acquisition (SCADA) system. Several synchrophasor-based algorithms and techniques have been and continue to be developed for real-time operation applications such as state estimation, stability analysis, disturbance detection, dynamic security assessment etc. However, synchrophasor data quality limits the incorporation of synchrophasor-based applications into control room operations environment and processes. The goal of this project is to develop methods that can improve synchrophasor data quality by recovering missing data reliably and efficiently. Data recovery refers to methods that estimate the values of missing data in the synchrophasor streams. Recently, modeless missing data recovery methods have been developed, that exploit the low-rank property of the spatial-temporal synchrophasor data blocks. A spatial-temporal synchrophasor data block can be considered as a matrix that is constructed by the measurements sampled at consecutive time instants with each row denoting the measurement of one certain channel across time. By exploiting the low-rank property of synchrophasor data matrices, the missing data recovery can be formulated as a low-rank matrix completion problem. The low-rank matrix completion problem has been extensively studied in the past few years and several algorithms have been developed to recover a low-rank matrix from partial observations. In this study, synchrophasor data analysis has been combined with low-rank matrix completion theory to develop a missing synchrophasor data recovery technique and tool.