Augmented Lagrange Multiplier Method Research Articles (Page 1)

Tensorized multi-dimensional multi-view clustering based on nonnegative matrix factorization.

ABSTRACTThis paper extends the energy‐based homogenization method (EBHM) to evaluate the effective elastic properties of two‐dimensional arc unit cells, where the periodic boundary conditions are constructed based on the polar strain–displacement relationships, and the displacement field is solved by the polar finite element method. Combining the extended EBHM with the parametric level set method (PLSM) and the augmented Lagrangian multiplier method (ALM), it is able to design the topological configurations of arc unit cells for seeking superior effective elastic properties. In addition, the level set function is first filtered by compactly supported radial basis functions (CSRBFs) in order to eliminate small holes in the configuration. Numerical examples are performed to demonstrate the advantages of the proposed method. The periodic boundary condition performs well when the circumferential angle of the arc unit cell is in the range of [0.0002, 0.2]. Results indicate that the effective elastic properties of optimized arc unit cells override those of geometrically transformed optimal square unit cells. This superiority mainly benefits from the fact that unisymmetrical configurations are allowed in arc unit cells but not in square unit cells. Results also imply that the capability of nucleation of new holes is retained even though small holes are eliminated by filtering. It contributes to the trade‐off between performance and manufacturability.

Extreme learning machine (ELM) has received increasingly more attention because of its high efficiency and ease of implementation. However, the existing ELM algorithms generally suffer from the drawbacks of noise sensitivity and poor robustness. Therefore, we combine the advantages of twin hyperplanes with the fast speed of ELM, and then introduce the characteristics of heteroscedastic Gaussian noise. In this paper, a new regressor is proposed, which is called twin extreme learning machine based on heteroskedastic Gaussian noise (TELM-HGN). In addition, the augmented Lagrange multiplier method is introduced to optimize and solve the presented model. Finally, a significant number of experiments were conducted on different data-sets including real wind-speed data, Boston housing price dataset and stock dataset. Experimental results show that the proposed algorithms not only inherits most of the merits of the original ELM, but also has more stable and reliable generalization performance and more accurate prediction results. These applications demonstrate the correctness and effectiveness of the proposed model.

We present a general framework for the analysis and modelling of frictional contact involving composite materials. The study has focused on composite materials formed by a matrix of rubber and synthetic or metallic fibres, which is the case of standard tires. We detail the numerical treatment of incompressibility at large deformations that rubber can experience, as well as the stiffening effect that properly oriented fibres will induce within the rubber. To solve the frictional contact between solids, a Dual Augmented Lagrangian Multiplier Method is used together with the Mortar method. This ensures a variationally consistent estimation of the contact forces. A modified Serial-Parallel Rule of Mixtures is employed to model the behaviour of composite materials. This is a simple and novel methodology that allows the blending of constitutive behaviours as diverse as rubber (very low stiffness and incompressible behaviour) and steel (high stiffness and compressible behaviour) taking into account the orientation of the fibres within the material. The locking due to the incompressibility constraint in the rubber material has been overcome by using Total Lagrangian mixed displacement-pressure elements. A collection of numerical examples is provided to show the accuracy and consistency of the methodology presented when solving frictional contact, incompressibility and composite materials under finite strains.

Hyperspectral image (HSI) is often corrupted by various kinds of noises. This letter proposes an innovative HSI denoising approach by leveraging the graph signal processing (GSP) theory and the low-rank (LR) matrix recovery model. With GSP, the piecewise smoothness (PWS) property of the HSI can be efficiently characterized, leading to a new regularization for HSI denoising, termed the adaptive weight graph total variation (AWGTV) regularization. Then, the denoising problem is formulated into a constrained optimization problem that incorporates the AWGTV and the LR property of the HSI. An augmented Lagrange multiplier method is adopted to solve the problem. Numerical experiments conducted on synthetic and real-world datasets and comparisons with existing methods demonstrate the effectiveness of the proposed denoising algorithm.

A class of bilinear matrix constraint optimization problem and its applications

To solve the energy‐efficient virtual network embedding problem, this study proposes an embedding algorithm based on Hopfield neural network. An energy‐efficient virtual network embedding model was established. Wavelet diffusion was performed to take the structural feature value into consideration and provide a candidate set for virtual network embedding. In addition, the Hopfield network was used in the candidate set to solve the virtual network energy‐efficient embedding problem. The augmented Lagrangian multiplier method was used to transform the energy‐efficient virtual network embedding constraint problem into an unconstrained problem. The resulting unconstrained problem was used as the energy function of the Hopfield network, and the network weight was iteratively trained. The energy‐efficient virtual network embedding scheme was obtained when the energy function was balanced. To prove the effectiveness of the proposed algorithm, we designed two experimental environments, namely, a medium‐sized scenario and a small‐sized scenario. Simulation results show that the proposed algorithm achieved a superior performance and effectively decreased the energy consumption relative to the other methods in both scenarios. Furthermore, the proposed algorithm reduced the number of open nodes and open links leading to a reduction in the overall power consumption of the virtual network embedding process, while ensuring the average acceptance ratio and the average ratio of the revenue and cost.

Foreground detection is one of the key steps in computer vision applications. Many foreground and background models have been proposed and achieved promising performance in static scenes. However, due to challenges such as dynamic background, irregular movement, and noise, most algorithms degrade sharply in complex scenes. To address the problem, we propose a motion-aware structured matrix factorization approach (MSMF), which integrates the structural and spatiotemporal motion information into a unified sparse-low-rank matrix factorization framework. Technologically, it has three main contributions: First, a variant of structured sparsity-inducing norm is proposed to constrain both structure and sparsity of foreground. The model is robust to the statistical variability of the underlying foreground pixels in complex scenes. Second, to capture the ambiguous pixels, a spatiotemporal cube-based motion trajectory is extracted for assisting matrix factorization. Finally, to solve the optimization problem of structured matrix factorization, we develop an augmented Lagrange multiplier method with the alternating direction strategy and Douglas-Rachford monotone operator splitting algorithm. Experiments demonstrate that the proposed approach achieves impressive performance in separating irregular moving foreground while suppressing the dynamic background and the noise, and outperforms some state-of-the-art algorithms.

Hyperspectral image denoising by total variation-regularized bilinear factorization

The standard traffic assignment problem (TAP) is often augmented with additional constraints to address non-standard applications. These models are called TAP with side constraints (TAPSC). Despite the rising significance of TAPSC models, the ability to efficiently solve them to satisfactory precision remains limited in real-world applications. The purpose of this paper is to fill this gap by integrating a recently developed high performance TAP solver, known as the path-based Greedy algorithm, with the augmented Lagrangian multiplier (ALM) method. This paper examines how precisely the subproblems in the ALM method should be solved to optimize the overall convergence performance. It is found that insufficiently converged subproblem solutions sometimes lead to catastrophic failures, although pursuing extremely high precision could also be counterproductive. Accordingly, it is proposed to adjust the precision required to solve the subproblems based on an approximate gap measured by the augmented Lagrangian. Results of numerical experiments show that adaptively adjusting the subproblem precision limit produces a 25% speed-up compared with the algorithm with a fixed limit.

The sparse signal reconstruction of compressive sensing can be accomplished by $${l_1}$$-norm minimization, but in many existing algorithms, there are the problems of low success probability and high computational complexity. To overcome these problems, an algorithm based on the alternating direction method of multipliers is proposed. First, using variable splitting techniques, an additional variable is introduced, which is tied to the original variable via an affine constraint. Then, the problem is transformed into a non-constrained optimization problem by means of the augmented Lagrangian multiplier method, where the multipliers can be obtained using the gradient ascent method according to dual optimization theory. The $${l_1}$$-norm minimization can finally be solved by cyclic iteration with concise form, where the solution of the original variable could be obtained by a projection operator, and the auxiliary variable could be solved by a soft threshold operator. Simulation results show that a higher signal reconstruction success probability is obtained when compared to existing methods, while a low computational cost is required.

Hyperspectral video sequences (HVSs) are well suited for gas plume detection (GPD). The high spectral resolution allows the detection of chemical clouds even when they are optically thin. Processing this new type of video sequences is challenging and requires advanced image and video analysis algorithms. In this paper, we propose a novel method for GPD recorded in HVSs. Based on the assumption that the background is stationary and the gas plume is moving, the proposed method separates the background from the gas plume via a low-rank and sparse decomposition. Furthermore, taking into consideration that the gas plume is continuous in both spatial and temporal dimensions, we include total variation regularization in the constrained minimization problem, which we solve using the augmented Lagrangian multiplier method. After applying the above process to each extracted feature, a novel fusion strategy is proposed to combine the information into a final detection result. Experimental results using real data sets indicate that the proposed method achieves very promising GPD performance.

Principal Component Analysis (PCA) as a tool for dimensionality reduction is widely used in many areas. In the area of bioinformatics, each involved variable corresponds to a specific gene. In order to improve the robustness of PCA-based method, this paper proposes a novel graph-Laplacian PCA algorithm by adopting L1/2 constraint (L1/2 gLPCA) on error function for feature (gene) extraction. The error function based on L1/2-norm helps to reduce the influence of outliers and noise. Augmented Lagrange Multipliers (ALM) method is applied to solve the subproblem. This method gets better results in feature extraction than other state-of-the-art PCA-based methods. Extensive experimental results on simulation data and gene expression data sets demonstrate that our method can get higher identification accuracies than others.

A topology optimization methodology is proposed for the flexible multibody system undergoing both large overall motion and large deformation. The system of concern is modeled via the absolute nodal coordinate formulation. The equivalent static load method is employed to transform the topology optimization of the nonlinear dynamic response of the system into a static one, and evaluated to adapt to the absolute nodal coordinate formulation by splitting the elastic deformations of the flexible components from the overall motions of those components. During the static topology optimization, the material interface is implicitly described as the zero level set of a higher-dimensional scalar function. Then, the semi-implicit level set method with the additive operator splitting algorithm is employed to solve the corresponding Hamilton-Jacobi partial differential equation. In addition, the expert evaluation method of weights based on the grey theory is utilized to define the objective function, and a modified augmented Lagrange multiplier method is proposed to treat the inequality volume constraint so as to avoid the oscillation and drift of the volume. Finally, two numerical examples are provided to validate the proposed methodology.

Speckle noise removal via nonlocal low-rank regularization

In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis that is used to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary adaptation problem, in which the dictionary is adaptive to one signal rather than a training set in dictionary learning. This dictionary adaptation problem is solved by using the augmented Lagrangian multiplier (ALM) method iteratively. We further accelerate the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.

The restoration of blurred images corrupted by Poisson noise is an important task in various applications such as medical imaging, microscopy imaging, and so on. We focus on mean curvature-based regularization to address the Poisson noise image restoration problem. Furthermore, we derive a numerical algorithm based on the augmented Lagrange multiplier method with a splitting technique. In order to simultaneously demonstrate the effectiveness of the proposed method for Poisson noise removal with deblurring, we conduct systematic experiments on both nature images and biological images. Experimental results show that the proposed approach can produce higher quality results and more natural images compared to some state-of-the-art variational algorithms recently developed.

Lightness biased cartoon-and-texture decomposition for textile image segmentation

A crucial step in many manifold clustering and embedding algorithms involves selecting the neighbors of a data point. In this paper, a non-negative representation learning algorithm is proposed to select the neighbors of a data point, so that they are almost always lying in the tangent space of the manifold at the given query point. This is very important to multi-manifold clustering and manifold embedding. Convex geometry theory states that a data point on a face of a convex polytope can be a convex combination of the vertices of the polytope, and that the non-zero combination scalars correspond to the vertices of the face the point lies on. The intent of this paper is to use points that that are near to a point as the vertices of a convex polytope, and select the vertices on the same face as this point to be the neighbors. Clearly, the point can be a convex combination of its neighbors. We can observe that these neighbors almost always lie in the tangent space of the manifold at the query point and can preserve the local manifold structure well. Based on this basic idea, we construct an objective function. Moreover, an augmented Lagrange multipliers method is proposed for solving it to derive a non-negative representation, and then the neighbors of a data point can be obtained. The result of our neighbor selection algorithm can be used by other clustering methods such as LEM or manifold embedding methods such as LLE. We demonstrate the effectiveness and efficiency of the proposed method through experiments on synthetic data and a real-world problem.

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problem of matrix completion and robust principal component analysis, namely recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries and from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, we consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the -norm in this paper. Meanwhile, we put forward a novel and effective algorithm called augmented Lagrange multipliers to exactly solve the problem. For mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy image following an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality performance of our method is dominant if images possess low-rank features such as geometrically regular textures and similar structured contents; especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and the restoration ability for a low-rank image matrix, but also in the preservation of textures and details in the image.