Low-rank Matrix Research Articles

Robust fault detection for chemical processes based on dynamic low-rank matrix and optimized LSTM

The data collected by sensors in modern chemical process systems are always contaminated by industrial noise, so robust fault detection is an important technology to ensure process safety. However, the existing robust detection techniques have limitations in extracting dynamic and time series features of chemical process data. In this paper, a robust fault detection method based on dynamic low-rank matrix and optimized LSTM is proposed for dynamic chemical processes under noise background. First, a new low-rank matrix decomposition method, dynamic principal component pursuit (DPCP), is proposed for the dynamic characteristics of process data containing noise. The PCP is improved using the time delay excursion method, i.e., the DPCP is constructed by embedding an augmented dynamic matrix with a time lag factor on the PCP and solved using the alternating direction multiplier method. The purpose is to strip the useless noise from the useful detection information and extract the dynamic low-rank matrix. Second, an optimized LSTM (OPLSTM) is proposed for the time series features of dynamic low-rank matrix. vector factors for feature selection of memory cells in the LSTM are reweighted to balance the time series features of industrial processes and avoid heavy reliance on erroneous fault information. In addition, support vector data description (SVDD) is used to describe the hypersphere with clear boundary, and distance-based statistic is constructed to achieve fault detection with high fault detection rate and low false alarm rate. Finally, to assess the effectiveness of the proposed method, we performed extensive experiments on the Tennessee Eastman Process and the electrolytic aluminum process, and compared them with multiple methods. The experimental results show that the proposed method is effective and robust for chemical processes fault detection in the background of industrial noise.

A nonlinear multi-label learning model based on Tanh mapping

The relationship between features and labels plays an important role in multi-label learning. The purpose of multi-label learning is to learn a mapping from the feature space to the label space. Many of the existing methods decompose the observed label matrix into the product of a low-rank latent label matrix and a projection matrix, and construct a linear mapping between the feature space and low-rank latent label space. However, in practice, the relationship between two high-dimensional spaces is often nonlinear. Consequently, these linear models may not sufficiently capture the data distribution and the true dependencies between features and labels. Herein, a novel classification model for multi-label learning is proposed, which considers nonlinear mapping from the feature space to the latent label space. Because of correlation between labels, the label matrix is usually of a low rank. To explore the low-rank structure, a latent label matrix is obtained using the decomposition of the original label matrix. To characterize the nonlinear relationship between features and latent labels, the tanh function is employed to learn a nonlinear classifier. The tanh function describes the distribution of labels by constraining the domain of the latent label space to the interval [−1,1]. In addition, the Frobenius norm regularization constraint is imposed on the variables in the model for optimization. The experimental results demonstrate that the proposed method is robust and comparable to state-of-the-art multi-label learning methods.

Structurally incoherent adaptive weighted low-rank matrix decomposition for image classification

Structurally incoherent adaptive weighted low-rank matrix decomposition for image classification

Effective elastic properties of three-dimensional multiple crack problems with the isogeometric boundary element parallel fast direct solver

We propose an isogeometric boundary element (IGABEM) parallel fast direct solver based on OpenMP to solve three-dimensional multiple crack problems. In this paper, the orderly and randomly distributed cracks are studied to observe the influence of the multiple crack interaction on the effective elastic properties. This paper designs the parallel implementation of the matrix low rank approximation and the matrix assembly for the hierarchical off-diagonal low rank (HODLR) matrix. In the matrix low rank approximation, we propose the m-n tree to improve the accelerated hybrid algorithm to balance the CPU performance and memory usage. The numerical results show that the effective elastic properties can respond to the strong interaction to a certain extent, but the maximum stress intensity factors (SIFs) of multiple cracks still need to be calculated to diagnose the possible crack initiation points. On the other hand, the numerical results also show that the IGABEM parallel fast direct solver can greatly improve the computational efficiency on the premise of ensuring the computational accuracy.

Matrix Completion With Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling

While uniform sampling has been widely studied in the matrix completion literature, CUR sampling approximates a low-rank matrix via row and column samples. Unfortunately, both sampling models lack flexibility for various circumstances in real-world applications. In this work, we propose a novel and easy-to-implement sampling strategy, coined Cross-Concentrated Sampling (CCS). By bridging uniform sampling and CUR sampling, CCS provides extra flexibility that can potentially save sampling costs in applications. In addition, we also provide a sufficient condition for CCS-based matrix completion. Moreover, we propose a highly efficient non-convex algorithm, termed Iterative CUR Completion (ICURC), for the proposed CCS model. Numerical experiments verify the empirical advantages of CCS and ICURC against uniform sampling and its baseline algorithms, on both synthetic and real-world datasets.

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.

Robust and Energy Efficient Sparse-Coded OFDM-DCSK System via Matrix Recovery

In this paper, we devise a sparse-coded orthogonal frequency division multiplexing (OFDM) differential chaos shift keying (DCSK) communication system based on low-rank matrix recovery which can handle Gaussian background noise and outlier-contaminated symbols simultaneously. As the noise-free OFDM-DCSK symbol matrix has rank 1, we exploit the vector outer product for its modeling, while sparse coding is also applied to reduce the transmission energy. To demodulate information bits from the sparse-coded signal, we formulate an objective function which consists of a sum of Frobenius norm for rank-1 matrix recovery and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm for identifying the possibly outlier-contaminated symbols, with a self-adaptive weight parameter. The resultant optimization problem is solved iteratively via block coordinate descent, and the Laplacian kernel with the Silverman’s rule is adopted for outlier detection. Theoretical analysis including convergence of the objective function, bit error rate (BER), energy efficiency and computational complexity, are provided. Simulation results show that the proposed system has comparable mean square error and BER performance with the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub> -norm minimization based matrix recovery approach at <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> = 2 in additive white Gaussian noise, and is superior to that of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> = 1 in Middleton class A noise, even when sparse coding is applied. Moreover, compared with other binary DCSK systems, our system achieves higher energy efficiency thanks to the sparse coding.

Affine Subspace Robust Low-Rank Self-Representation: from Matrix to Tensor.

Low-rank self-representation based subspace learning has confirmed its great effectiveness in a broad range of applications. Nevertheless, existing studies mainly focus on exploring the global linear subspace structure, and cannot commendably handle the case where the samples approximately (i.e., the samples contain data errors) lie in several more general affine subspaces. To overcome this drawback, in this paper, we innovatively propose to introduce affine and nonnegative constraints into low-rank self-representation learning. While simple enough, we provide their underlying theoretical insight from a geometric perspective. The union of two constraints geometrically restricts each sample to be expressed as a convex combination of other samples in the same subspace. In this way, when exploring the global affine subspace structure, we can also consider the specific local distribution of data in each subspace. To comprehensively demonstrate the benefits of introducing two constraints, we instantiate three low-rank self-representation methods ranging from single-view low-rank matrix learning to multi-view low-rank tensor learning. We carefully design the solution algorithms to efficiently optimize the proposed three approaches. Extensive experiments are conducted on three typical tasks, including single-view subspace clustering, multi-view subspace clustering, and multi-view semi-supervised classification. The notably superior experimental results powerfully verify the effectiveness of our proposals.

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.

Hybrid Threshold Denoising Framework Using Singular Value Decomposition for Side-Channel Analysis Preprocessing.

The traces used in side-channel analysis are essential to breaking the key of encryption and the signal quality greatly affects the correct rate of key guessing. Therefore, the preprocessing of side-channel traces plays an important role in side-channel analysis. The process of side-channel leakage signal acquisition is usually affected by internal circuit noise, external environmental noise, and other factors, so the collected signal is often mixed with strong noise. In order to extract the feature information of side-channel signals from very low signal-to-noise ratio traces, a hybrid threshold denoising framework using singular value decomposition is proposed for side-channel analysis preprocessing. This framework is based on singular value decomposition and introduces low-rank matrix approximation theory to improve the rank selection methods of singular value decomposition. This paper combines the hard threshold method of truncated singular value decomposition with the soft threshold method of singular value shrinkage damping and proposes a hybrid threshold denoising framework using singular value decomposition for the data preprocessing step of side-channel analysis as a general preprocessing method for non-profiled side-channel analysis. The data used in the experimental evaluation are from the raw traces of the public database of DPA contest V2 and AES_HD. The success rate curve of non-profiled side-channel analysis further confirms the effectiveness of the proposed framework. Moreover, the signal-to-noise ratio of traces is significantly improved after preprocessing, and the correlation with the correct key is also significantly enhanced. Experimental results on DPA v2 and AES_HD show that the proposed noise reduction framework can be effectively applied to the side-channel analysis preprocessing step, and can successfully improve the signal-to-noise ratio of the traces and the attack efficiency.

Speech Enhancement Based on Constrained Low-rank Sparse Matrix Decomposition Integrated with Temporal Continuity Regularisation

Speech enhancement in strong noise condition is a challenging problem. Low-rank and sparse matrix decomposition (LSMD) theory has been applied to speech enhancement recently and good performance was obtained. Existing LSMD algorithms consider each frame as an individual observation. However, real-world speeches usually have a temporal structure, and their acoustic characteristics vary slowly as a function of time. In this paper, we propose a temporal continuity constrained low-rank sparse matrix decomposition (TCCLSMD) based speech enhancement method. In this method, speech separation is formulated as a TCCLSMD problem and temporal continuity constraints are imposed in the LSMD process. We develop an alternative optimisation algorithm for noisy spectrogram decomposition. By means of TCCLSMD, the recovery speech spectrogram is more consistent with the structure of the clean speech spectrogram, and it can lead to more stable and reasonable results than the existing LSMD algorithm. Experiments with various types of noises show the proposed algorithm can achieve a better performance than traditional speech enhancement algorithms, in terms of yielding less residual noise and lower speech distortion.

Pattern synthesis for symmetric linear transducer array by bilinear factorization of Hankel matrices

We investigate the problem of designing an unequally spaced symmetric linear array with a minimum number of elements. For unequally spaced arrays, simultaneously tuning all the elements' positions and weights leads to a non-convex problem. Inspired by the fact that the matrix pencil method (MPM) can estimate the weights and positions of array elements from a Hankel matrix formed by samples of beampattern, we attack the design problem by finding the Hankel matrix fulfilling specifications with the minimum rank. Fortunately, the low-rank optimization can be addressed by nuclear norm relaxation. In the proposed approach, we minimize the nuclear norm of the Hankel matrix utilizing the low-rank matrix factorization-based augmented Lagrangian alternating direction method. Once the low-rank Hankel matrix is obtained, we finish the array design with the help of MPM. By finding the desired Hankel matrix with a minimum nuclear norm, the proposed approach avoids the NP-hard problem of simultaneously optimizing the weights and positions of elements. Moreover, the proposed method does not need a reference beampattern and makes no assumptions on element positions, making it more flexible in facing real-world problems. We compare the proposed approach with existing methods under different scenarios and prove its effectiveness.

Topology identification under spatially correlated noise

This article addresses the problem of reconstructing the topology of a network of agents interacting via linear dynamics, while being excited by exogenous stochastic sources that are possibly correlated across the agents, from time-series measurements alone. It is shown, under the assumption that the correlations are affine in nature, such network of nodal interactions is equivalent to a network with added agents. The added agents are represented by nodes that are latent, where no corresponding time-series measurements are available; however, here all the exogenous excitements are spatially (that is, across agents) uncorrelated. Generalizing affine correlations, it is shown that, under polynomial correlations, the latent nodes in the expanded networks can be excited by clusters of noise sources, where the clusters are uncorrelated with each other. The clusters can be replaced with a single noise source if the latent nodes are allowed to have non-linear interactions. Finally, using the sparse plus low-rank matrix decomposition of the imaginary part of the inverse power spectral density matrix (IPSDM) of the time-series data, the topology of the network is reconstructed. Under non conservative assumptions, the correlation graph of the noise sources is retrieved.

Unsupervised dimensionality reduction of medical hyperspectral imagery in tensor space

Background and objectivesCompared with traditional RGB images, medical hyperspectral imagery (HSI) has numerous continuous narrow spectral bands, which can provide rich information for cancer diagnosis. However, the abundant spectral bands also contain a large amount of redundancy information and increase computational complexity. Thus, dimensionality reduction (DR) is essential in HSI analysis. All vector-based DR methods ignore the cubic nature of HSI resulting from vectorization. To overcome the disadvantage of vector-based DR methods, tensor-based techniques have been developed by employing multi-linear algebra. MethodsTo fully exploit the structure features of medical HSI and enhance computational efficiency, a novel method called unsupervised dimensionality reduction via tensor-based low-rank collaborative graph embedding (TLCGE) is proposed. TLCGE introduces entropy rate superpixel (ERS) segmentation algorithm to generate superpixels. Then, a low-rank collaborative graph weight matrix is constructed on each superpixel, greatly improving the efficiency and robustness of the proposed method. After that, TLCGE reduces dimensions in tensor space to well preserve intrinsic structure of HSI. ResultsThe proposed TLCGE is tested on cholangiocarcinoma microscopic hyperspectral data sets. To further demonstrate the effectiveness of the proposed algorithm, other machine learning DR methods are used for comparison. Experimental results on cholangiocarcinoma microscopic hyperspectral data sets validate the effectiveness of the proposed TLCGE. ConclusionsThe proposed TLCGE is a tensor-based DR method, which can maintain the intrinsic 3-D data structure of medical HSI. By imposing the low-rank and sparse constraints on the objective function, the proposed TLCGE can fully explore the local and global structures within each superpixel. The computational efficiency of the proposed TLCGE is better than other tensor-based DR methods, which can be used as a preprocessing step in real medical HSI classification or segmentation.

Low-rank matrix recovery via novel double nonconvex nonsmooth rank minimization with ADMM

Low-rank matrix recovery via novel double nonconvex nonsmooth rank minimization with ADMM

Saliency-Aided Online RPCA for Moving Target Detection in Infrared Maritime Scenarios.

Moving target detection (MTD) is a crucial task in computer vision applications. In this paper, we investigate the problem of detecting moving targets in infrared (IR) surveillance video sequences captured using a steady camera in a maritime setting. For this purpose, we employ robust principal component analysis (RPCA), which is an improvement of principal component analysis (PCA) that separates an input matrix into the following two matrices: a low-rank matrix that is representative, in our case study, of the slowly changing background, and a sparse matrix that is representative of the foreground. RPCA is usually implemented in a non-causal batch form. To pursue a real-time application, we tested an online implementation, which, unfortunately, was affected by the presence of the target in the scene during the initialization phase. Therefore, we improved the robustness by implementing a saliency-based strategy. The advantages offered by the resulting technique, which we called "saliency-aided online moving window RPCA" (S-OMW-RPCA) are the following: RPCA is implemented online; along with the temporal features exploited by RPCA, the spatial features are also taken into consideration by using a saliency filter; the results are robust against the condition of the scene during the initialization. Finally, we compare the performance of the proposed technique in terms of precision, recall, and execution time with that of an online RPCA, thus, showing the effectiveness of the saliency-based approach.

Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds

Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds

Multi-modal multi-step wind power forecasting based on stacking deep learning model

Wind power is becoming a clean and effective energy source for electric power generation. However, the abnormity, multi-modal, and uncertainty represented in wind power data are commonly undesired. Thus, accurate wind power forecasting is a significant method for keeping the power system operations steady. To solve these issues, a multi-modal multi-step wind power forecasting model is presented. To obtain this, the density-based spatial clustering of applications with noise (DBSCAN) is improved by the k-dimensional tree (kd-tree) for detecting abnormal data. Then, the low-rank matrix fusion method fuses the wind speed, wind direction, and air density modalities for obtaining a unified representation. To further increase model accuracy, we propose a stacking deep learning model (SDLM) for overcoming the uncertainty phenomenon, which contains the bidirectional gated recurrent unit (BGRU) and leaky echo state network (LESN). The final forecasting results are acquired by a meta-learning operator. To validate the accuracy and stability of the presented approach, the inland and offshore wind farm datasets are used for forecasting. The contrastive results demonstrate that the presented model outperforms satisfactory performance in multi-step wind power prediction.

Constant Wideband Compressive Spectrum Sensing for Sparse and Nonsparse Signals With Low-Rank Matrix Recovering and Prior Knowledge Mining

In cognitive radios (CR), compressive spectrum sensing (CSS) is a promising technique to detect wideband spectrum holes with reduced sampling rate requirement of analog-to-digital converter (ADC). It is implied in compressive sensing (CS) that the minimum sampling rate is determined by the sparsity order of the observed wideband signal. Unfortunately, the sparsity order is previously unknown and dynamically varies. Meanwhile, CS technology fails to work efficiently for the nonsparse spectrum case. For example, the CSS performance degenerates significantly for low signal-to-noise ratio (SNR) scenarios since the observed signal becomes nonsparse in frequency domain. To overcome the above-mentioned challenges, we propose a novel CSS algorithm with temporally-correlated prior knowledge mining (TPKM-CSS) to detect spectrum holes without a prior sparsity order information. The proposed scheme can handle both the sparse and nonsparse spectrum cases. The proposed solution has three contributions: firstly, CR implements multiple parallel sampling simultaneously to acquire wireless spectrum signals, which have a low-rank spectrum matrix. Secondly, the temporal correlation between continuous CS data is further exploited to improve the estimation accuracy of the current spectrum. Finally, the theoretical performance of the proposed TPKM-CSS is evaluated, wherein the cumulative distribution function (CDF) of the spectrum reconstruction error is derived in the closed form. The simulation results show that the proposed scheme works well for both sparse and nonsparse spectrum scenarios. Compared with the CSS schemes that are based on low-rank matrix recovery (LMR-CSS), maximizing correlation (MC), iterative compressive filtering (ICF), or Bayesian CS (BCS), the proposed scheme yields the superior performance in terms of the probability of detection and the probability of false alarm, especially for nonsparse signals. Furthermore, the simulated statistical performance of the spectrum reconstruction error matches well the theoretical CDF curve.