Low-rank Method Research Articles

Hyperspectral Anomaly Detection via Structured Sparsity Plus Enhanced Low-Rankness

Hyperspectral anomaly detection (HAD), distinguishing anomalous pixels or subpixels from the background, has received increasing attention in recent years. Low-Rank Representation (LRR)-based methods have also been promoted rapidly for HAD, but they may encounter three challenges: (1) they adopted the nuclear norm as the convex approximation, yet a sub-optimal solution of the rank function; (2) they overlook the structured spatial correlation of anomalous pixels; (3) they fail to comprehensively explore the local structure details of the original background. To address these challenges, in this paper, we proposed the Structured Sparsity Plus Enhanced Low-Rank (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ELR) method for HAD. Specifically, our S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ELR method adopts the weighted tensor Schatten- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> norm, acting as an enhanced approximation of the rank function than the tensor nuclear norm, and the structured sparse norm to characterize the low-rank properties of the background and the sparsity of the abnormal pixels, respectively. To preserve the local structural details, the position-based Laplace regularizer is accompanied. An iterative algorithm is derived from the popular alternating direction methods of multipliers. Compared to the existing state-of-the-art HAD methods, the experimental results have demonstrated the superiority of our proposed S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ELR method.

Low-Rank Room Impulse Response Estimation

In this paper we consider low-rank estimation of room impulse responses (RIRs). Inspired by a physics-driven room-acoustical model, we propose an estimator of RIRs that promotes a low-rank structure for a matricization, or reshaping, of the estimated RIR. This low-rank prior acts as a regularizer for the inverse problem of estimating an RIR from input-output observations, preventing overfitting and improving estimation accuracy. As directly enforcing a low rank of the estimate results is an NP-hard problem, we consider two different relaxations, one using the nuclear norm, and one using the recently introduced concept of quadratic envelopes. Both relaxations allow for implementing the proposed estimator using a first-order algorithm with convergence guarantees. When evaluated on both synthetic and recorded RIRs, it is shown that under noisy output conditions, or when the spectral excitation of the input signal is poor, the proposed estimator outperforms comparable existing methods. The performance of the two low-rank relaxations methods is similar, but the quadratic envelope has the benefit of superior robustness to the choice of regularization hyperparameter in the case when the signal-to-noise ratio is unknown. The performance of the proposed method is compared to that of ordinary least squares, Tikhonov least squares, as well as the Cramér-Rao lower bound (CRLB).

Regularized Low-Rank Approximation Method for Diffraction Enhancement

Regularized Low-Rank Approximation Method for Diffraction Enhancement

Efficient and Differentiable Low-Rank Matrix Completion With Back Propagation

The low-rank matrix completion has gained rapidly increasing attention from researchers in recent years for its efficient recovery of the matrix in various fields. Numerous studies have exploited the popular neural networks to yield low-rank outputs under the framework of low-rank matrix factorization. However, due to the discontinuity and nonconvexity of rank function, it is difficult to directly optimize the rank function via back propagation. Although a large number of studies have attempted to find relaxations of the rank function, e.g., Schatten- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> norm, they still face the following issues when updating parameters via back propagation: 1) These methods or surrogate functions are still non-differentiable, bringing obstacles to deriving the gradients of trainable variables. 2) Most of these surrogate functions perform singular value decomposition upon the original matrix at each iteration, which is time-consuming and blocks the propagation of gradients. To address these problems, in this paper, we develop an efficient block-wise model dubbed differentiable low-rank learning (DLRL) framework that adopts back propagation to optimize the Multi-Schatten- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> norm Surrogate (MSS) function. Distinct from the original optimization of this surrogate function, the proposed framework avoids singular value decomposition to admit the gradient propagation and builds a block-wise learning scheme to minimize values of Schatten- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> norms. Accordingly, it speeds up the computation and makes all parameters in the proposed framework learnable according to a predefined loss function. Finally, we conduct substantial experiments in terms of image recovery and collaborative filtering. The experimental results verify the superiority of the proposed framework in terms of both runtimes and learning performance compared with other state-of-the-art low-rank optimization methods. Our codes are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/chenzl23/DLRL</uri> .

Computation of the self-diffusion coefficient with low-rank tensor methods: application to the simulation of a cross-diffusion system

Cross-diffusion systems arise as hydrodynamic limits of lattice multi-species interacting particle models. The objective of this work is to provide a numerical scheme for the simulation of the cross-diffusion system identified in [J. Quastel, Comm. Pure Appl. Math., 45 (1992), pp. 623–679]. To simulate this system, it is necessary to provide an approximation of the so-called self-diffusion coefficient matrix of the tagged particle process. Classical algorithms for the computation of this matrix are based on the estimation of the long-time limit of the average mean square displacement of the particle. In this work, as an alternative, we propose a novel approach for computing the self-diffusion coefficient using deterministic low-rank approximation techniques, as the minimum of a high-dimensional optimization problem. The computed self-diffusion coefficient is then used for the simulation of the cross-diffusion system using an implicit finite volume scheme.

Face Recognition Method under Adaptive Image Matching and Dictionary Learning Algorithm.

In this research, a robust face recognition method based on adaptive image matching and a dictionary learning algorithm was proposed. A Fisher discriminant constraint was introduced into the dictionary learning algorithm program so that the dictionary had certain category discrimination ability. The purpose was to use this technology to reduce the influence of pollution, absence, and other factors on face recognition and improve the recognition rate. The optimization method was used to solve the loop iteration to obtain the expected specific dictionary, and the selected specific dictionary was used as the representation dictionary in adaptive sparse representation. In addition, if a specific dictionary was placed in a seed space of the original training data, the mapping matrix can be used to represent the mapping relationship between the specific dictionary and the original training sample, and the test sample could be corrected according to the mapping matrix to remove the contamination in the test sample. Moreover, the feature face method and dimension reduction method were used to process the specific dictionary and the corrected test sample, and the dimensions were reduced to 25, 50, 75, 100, 125, and 150, respectively. In this research, the recognition rate of the algorithm in 50 dimensions was lower than that of the discriminatory low-rank representation method (DLRR), and the recognition rate in other dimensions was the highest. The adaptive image matching classifier was used for classification and recognition. The experimental results showed that the proposed algorithm had a good recognition rate and good robustness against noise, pollution, and occlusion. Health condition prediction based on face recognition technology has the advantages of being noninvasive and convenient operation.

Learning Tensor Low-Rank Representation for Hyperspectral Anomaly Detection.

Recently, low-rank representation (LRR) methods have been widely applied for hyperspectral anomaly detection, due to their potentials in separating the backgrounds and anomalies. However, existing LRR models generally convert 3-D hyperspectral images (HSIs) into 2-D matrices, inevitably leading to the destruction of intrinsic 3-D structure properties in HSIs. To this end, we propose a novel tensor low-rank and sparse representation (TLRSR) method for hyperspectral anomaly detection. A 3-D TLR model is expanded to separate the LR background part represented by a tensorial background dictionary and corresponding coefficients. This representation characterizes the multiple subspace property of the complex LR background. Based on the weighted tensor nuclear norm and the LF,1 sparse norm, a dictionary is designed to make its atoms more relevant to the background. Moreover, a principal component analysis (PCA) method can be assigned as one preprocessing step to exact a subset of HSI bands, retaining enough the HSI object information and reducing computational time of the postprocessing tensorial operations. The proposed model is efficiently solved by the well-designed alternating direction method of multipliers (ADMMs). A comparison with the existing algorithms via experiments establishes the competitiveness of the proposed method with the state-of-the-art competitors in the hyperspectral anomaly detection task.

Sparse Low-Rank Matrix Estimation With Nonconvex Enhancement for Fault Diagnosis of Rolling Bearings

The diagnosis of bearing early fault is significant and fundamental in the machine condition monitoring. The accurate and effective diagnosis is of great importance to avoid further serious accidents. However, existing sparse low-rank (SLR) methods for bearing fault diagnosis suffer from underestimation of amplitude and inaccurate approximation of singular values. Therefore, in this paper, a novel sparse low-rank matrix estimation method with nonconvex enhancement (SLRNE) is proposed, extracting the fault transients from observed noisy signal. Specifically, fault transients have both sparse and low-rank properties in time-frequency domain. Based on this, a SLR optimization model is proposed to simultaneously promote the above two properties via truncated nuclear norm (TNN) and generalized minimax concave (GMC) penalty function. The two nonconvex functions aim to promote low-rank property and sparsity respectively. Then, based on derived convexity conditions of the optimization problem, convex optimization algorithm, alternating direction method of multipliers (ADMM) and forward-and-backward splitting (FBS) algorithm, are applied to obtain global optimal solution. In the iterative algorithm, a weighting strategy is designed for the singular value threshold operator to enhance the effect of fault feature extraction. Simulated and experimental signals verify the effectiveness of SLRNE and contrast experiments verify its superiority.

Edge Preserved Low-Rank SAR Image Despeckling via Hierarchical Prior Knowledge Regulation

Synthetic aperture radar (SAR) image despeckling is a challenging task as speckle noise is spatially correlated and signal-dependent, and appears as a grainy texture superimposed on images. Although traditional low-rank SAR image despeckling methods have shown promising performance, they have the problem of producing over-smoothed images with blurred edges due to their low-rank characteristics. In this paper, we propose a novel edge preserved SAR despeckling method named EP-LRSID, which can keep rich edge details while reducing speckle noise. Specifically, EP-LRSID takes a fresh look at the low-rank model, i.e., we can obtain structural edge information from residuals which is viewed as noise and simply disregarded by the traditional low-rank methods. To obtain discriminative edge information from residuals, the edge subspace is obtained in a manifold framework by using a dynamic affinity graph regularization. Moreover, a new hierarchical prior knowledge regulation is designed to make different kinds of pixels processed hierarchically, especially the strong scattering points in SAR images. By introducing this prior knowledge, our low-rank model can obtain more confidential low-rank parts and edge parts, thus structural information including edges can be better preserved in this way. Extensive experiments on several real and synthetic datasets demonstrate that EP-LRSID can achieve the highest despeckling performance with edge preservation than other state-of-the-art despeckling algorithms.

Asymmetry total variation and framelet regularized nonconvex low-rank tensor completion

The low-rank tensor representation has shown enormous potential and advantages in diverse applications, such as image inpainting, image restoration. However, (1) due to the inherent limitations of the low-rank tensor model, the tensor nuclear norm is utilized yet as a biased estimate of the tensor rank, which limits the image recovery performance. (2) most current low-rank approximation-based methods focus on the preservation of global information, while ignoring the local details preservation such as the spatial piece-wise smoothness and sharp edges. To overcome the above obstacles, we propose a novel asymmetry three-dimensional total variation and framelet regularized nonconvex low-rank tensor completion (ATV-FNTC) model, which integrates the nonconvex penalty function and asymmetric three-dimensional total variation into one unified model. Specifically, ATV-FNTC introduces one nonconvex penalty function as a tighter regularizer to approximate the tensor rank. Different from existing methods, the asymmetric three-dimensional total variation (ATV) regularization is developed to achieve more accurate recovery of detailed information and flexibly control the smoothing strength of different dimensions of tensor data. Furthermore, we design an iterative algorithm to solve the nonconvex ATV-FNTC model by the alternating direction methods of multipliers. Experimental results reveal the superiority of the proposed method compared with several state-of-the-art methods when handling different completion tasks.

Hybrid Multimodal Medical Image Fusion Method Based on LatLRR and ED-D2GAN

In order to better preserve the anatomical structure information of Computed Tomography (CT) source images and highlight the metabolic information of lesion regions in Positron Emission Tomography (PET) source images, a hybrid multimodal medical image fusion method (LatLRR-GAN) based on Latent low-rank representation (LatLRR) and the dual discriminators Generative Adversarial Network (ED-D2GAN) is proposed. Firstly, considering the denoising capability of LatLRR, source images were decomposed by LatLRR. Secondly, the ED-D2GAN model was put forward as the low-rank region fusion method, which can fully extract the information contained by the low-rank region images. Among them, encoder and decoder networks were used in the generator; convolutional neural networks were also used in dual discriminators. Thirdly, a threshold adaptive weighting algorithm based on the region energy ratio is proposed as the salient region fusion rule, which can improve the overall sharpness of the fused image. The experimental results show that compared with the best methods of the other six methods, this paper is effective in multiple objective evaluation metrics, including the average gradient, edge intensity, information entropy, spatial frequency and standard deviation. The results of the two experiments are improved by 35.03%, 42.42%, 4.66%, 8.59% and 11.49% on average.

Low-rank tensor methods for Markov chains with applications to tumor progression models

Cancer progression can be described by continuous-time Markov chains whose state space grows exponentially in the number of somatic mutations. The age of a tumor at diagnosis is typically unknown. Therefore, the quantity of interest is the time-marginal distribution over all possible genotypes of tumors, defined as the transient distribution integrated over an exponentially distributed observation time. It can be obtained as the solution of a large linear system. However, the sheer size of this system renders classical solvers infeasible. We consider Markov chains whose transition rates are separable functions, allowing for an efficient low-rank tensor representation of the linear system’s operator. Thus we can reduce the computational complexity from exponential to linear. We derive a convergent iterative method using low-rank formats whose result satisfies the normalization constraint of a distribution. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank.

The legitimacy of decoupled dynamic flow stress equations and their representation based on discrete experimental data

In this study, the legitimacy of a decoupled empirical dynamic flow stress equations, typically the type of Johnson-Cook (J-C) (flow stress) equations, is mathematically verified and a criterion of J-C type equation is provided. The mechanical experimental data are assembled as 2D matrix (two variables of strain and strain-rate or temperature) or 3D array (three variables of strain, strain-rate and temperature). With low-rank approximation method, i.e. singular value decomposition (SVD) for 2D matrix and CANDECOMP/PARAFAC (CP) for 3D array, the experimental data matrix/array (N) can be decomposed into a heavily-weighted matrix/array (N1) and several matrices/arrays with reduced weights (N2, N3, …). The criterion of J-C type equation is that N1 can approximately represent N with acceptable error. Otherwise, the use of J-C type equation is invalid. A method to describe the dynamic flow stress based on discrete experimental dataset is further proposed based on the SVD/CP method. The advantages of coupled and decoupled flow stress equations and the problems associated with the original J-C equation are discussed.

MM-LMF: A Low-Rank Multimodal Fusion Dangerous Driving Behavior Recognition Method Based on FMCW Signals

Multimodal research is an emerging field of artificial intelligence, and the analysis of dangerous driving behavior is one of the main application scenarios in the field of multimodal fusion. Aiming at the problem of data heterogeneity in the process of behavior classification by multimodal fusion, this paper proposes a low-rank multimodal data fusion method, which utilizes the complementarity between data modalities of different dimensions in order to classify and identify dangerous driving behaviors. This method uses tensor difference matrix data to force low-rank fusion representation, improves the verification efficiency of dangerous driving behaviors through multi-level abstract tensor representation, and solves the problem of output data complexity. A recurrent network based on the attention mechanism, AR-GRU, updates the network input parameter state and learns the weight parameters through its gated structure. This model improves the dynamic connection between modalities on heterogeneous threads and reduces computational complexity. Under low-rank conditions, it can quickly and accurately classify and identify dangerous driving behaviors and give early warnings. Through a large number of experiments, the accuracy of this method is improved by an average of 1.76% compared with the BiLSTM method and the BiGRU-IAAN method in the training and verification of the self-built dataset.

A Single-Image Noise Estimation Algorithm Based on Pixel-Level Low-Rank Low-Texture Patch and Principal Component Analysis.

Noise level is an important parameter for image denoising in many image-processing applications. We propose a noise estimation algorithm based on pixel-level low-rank, low-texture subblocks and principal component analysis for white Gaussian noise. First, an adaptive clustering algorithm, based on a dichotomy merge, adaptive pixel-level low-rank matrix construction method and a gradient covariance low-texture subblock selection method, is proposed to construct a pixel-level low-rank, low-texture subblock matrix. The adaptive clustering algorithm can improve the low-rank property of the constructed matrix and reduce the content of the image information in the eigenvalues of the matrix. Then, an eigenvalue selection method is proposed to eliminate matrix eigenvalues representing the image to avoid an inaccurate estimation of the noise level caused by using the minimum eigenvalue. The experimental results show that, compared with existing state-of-the-art methods, our proposed algorithm has, in most cases, the highest accuracy and robustness of noise level estimation for various scenarios with different noise levels, especially when the noise is high.

Amplitude behavior in low-rank acoustic modeling

The numerical simulation of wave propagation can be represented by a propagator matrix applied to previous instances of the wavefield. Using the sparsity of the propagator matrix to approximate it by a low-rank representation, one can increment the wavefield’s phase from one time instance to another. Though this procedure does not pay attention to the amplitudes of the seismic waves, it is important to understand its dynamic properties. Here, we evaluate the amplitudes obtained by the low-rank method in the simulation of 2D acoustic wave propagation. In homogeneous media, where theoretical expressions for the wavefield are available, the method provides not only an excellent kinematic approximation, but also reliable amplitudes. For a single horizontal reflector below a homogeneous overburden, the reflection coefficients approximated by the low-rank method are of the same quality or slightly superior to those obtained by a second-order finite-difference (FD) method (implementation from SU). However, in more generally inhomogeneous media, our tests showed larger discrepancies between FD and low-rank modeling results. While comparing unfavorably with FD regarding computation time for small models, its quasi-linear scaling with model size makes the low-rank method superior for large models. Moreover, a generalization to more complex, e.g., anisotropic, media is straightforward.

Combining Deep Denoiser and Low-rank Priors for Infrared Small Target Detection

Many existing low-rank methods have achieved good detection performance in uniform scenes, but they suffer from a high false alarm rate in complex noisy scenes. Therefore, it is important to improve the detection performance of low-rank models in noisy scenes. In this paper, we first formulate an implicit regularizer by plugging a denoising neural network (termed as deep denoiser), which can learn deep image priors from a large number of natural images. Then, we use the weighted sum of weighted tensor nuclear norm for more accurate background estimation. Finally, alternating direction multiplier method is used to solve the model under the plug-and-play framework. By integrating low-rank prior with deep denoiser prior, our model achieves higher accuracy. Experiments on different scenes demonstrate that our method achieves an improved performance in terms of visual effects and quantitative metrics. Specially, the overall accuracy of AUC value (AUCOA) achieved by the proposed method on Sequences 1-6 are 1.24%, 1.16%, 0.63%, 1.9%, 0.82%, 2.06% higher than those achieved by the second top performing methods, respectively.

A sweep-based low-rank method for the discrete ordinate transport equation

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via the “unconventional” basis update & Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

Hypercomplex Low Rank Reconstruction for NMR Spectroscopy

Nuclear magnetic resonance (NMR) spectroscopy serves as an important tool to analyze chemicals and proteins in bioengineering. Multi-dimensional NMR offers a major improvement in resolution with multi-dimensional spectrum, but significantly increases data acquisition time and produces hypercomplex data that is difficult to be handled. To reduce this time, non-uniformly sampling can be applied to obtain undersampled data and using a reconstruction approach, such as the state-of-the-art low rank method, to remove the spectral artifacts introduced by undersampling. However, only complex format of signal, including the real and imaginary parts, is considered in previous low rank approach, which is less efficient when dealing with hypercomplex data that has multiple components. To solve this problem, a hypercomplex low rank model is proposed by introducing an adjoint matrix operation and then solved with a fast matrix factorization algorithm. This method explores redundant information among all the components of hypercomplex signal. Using simulated data and real protein data, we demonstrate that the proposed method provides a fast and high-fidelity reconstruction of hypercomplex spectroscopy in fast NMR.

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems.