Augmented Lagrangian Alternating Direction Method Research Articles

Reservation and allocation model considering the user cost and the utilization of parking space

In order to alleviate the problem of imbalance between parking supply and demand by improving the utilization rate of parking resources, and considering the game relationship between user benefits and system benefits in parking allocation, the optimization objective function of maximum parking space utilization and minimum user cost in parking reservation mode is established from the two aspects which are system optimization and user optimization respectively. Then, the optimal allocation parking integer programming model (OPA) considering users' preferences is established. An Augmented Lagrangian-Alternating Direction Method of Multipliers Algorithm is designed to solve the optimal solution of the model. Finally, in order to test the validity of the model, the proposed model and the classical distribution model are compared and analyzed under different supply and demand conditions. The results show that the performance of the proposed model in three performance metrics which are parking utilization, average user cost and request acceptance rate is significantly better than that of the classical allocation model. The research results of this paper can provide theoretical reference for the management of parking reservation platform.

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.

Low-Rank Tensor Regularized Fuzzy Clustering for Multiview Data

Since data are collected from a range of sources via different techniques, multiview clustering has become an emerging technique for unsupervised data classification. However, most existing soft multiview clustering methods only consider the pairwise correlations and ignore high-order correlations among multiple views. To integrate more comprehensive information from different views, this article innovates a fuzzy clustering model using the low-rank tensor to address the multiview data clustering problem. Our method first conducts a standard fuzzy clustering on different views of the data separately. Then, the obtained soft partition results are aggregated as the new data to be handled by a Kullback-Leibler (KL) divergence-based fuzzy model with low-rank tensor constraints. The KL divergence function, which replaces the traditional minimized Euclidean distance, can enhance the robustness of the model. More importantly, we formulate fuzzy partition matrices of different views as a third-order tensor. So, a low-rank tensor is introduced as a norm constraint in the KL divergence-based fuzzy clustering to obtain dexterously high-order correlations of different views. The minimization of the final model is convex and we present an efficient augmented Lagrangian alternating direction method to handle this problem. Specially, the global membership is derived by using tensor factorization. The efficiency and superiority of the proposed approach are demonstrated by the comparison with state-of-the-art multiview clustering algorithms on many multiple-view data sets.

Low-Rank Tensor Completion and Total Variation Minimization for Color Image Inpainting

Low-rank (LR) and total variation (TV) are two most frequent priors that occur in image processing problems, and they have sparked a tremendous amount of researches, particularly for moving from scalar to vector, matrix or even high-order based functions. However, discretization schemes used for TV regularization often ignore the difference of the intrinsic properties, so it will lead to the problem that local smoothness cannot be effectively generated, let alone the problem of blurred edges. To address the image inpainting problem with corrupted data, in this paper, the color images are naturally considered as three-dimensional tensors, whose prior of smoothness can be measured by varietal TV norm along different dimensions. Specifically, we propose incorporating Shannon total variation (STV) and low-rank tensor completion (LRTC) into the construction of the final cost function, in which a new nonconvex low-rank constraint, namely truncated $\gamma $ -norm, is involved for closer rank approximation. Moreover, two methods are developed, i.e., LRRSTV and LRRSTV-T, due to the fact that LRTC can be represented by tensor unfolding and tensor decomposition. The final solution can be achieved by a practical variant of the augmented Lagrangian alternating direction method (ALADM). Experiments on color image inpainting tasks demonstrate that the proposed methods perform better then the state-of-the-art algorithms, both qualitatively and quantitatively.

An augmented Lagrangian alternating direction method for overlapping community detection based on symmetric nonnegative matrix factorization

In this paper, we present an augmented Lagrangian alternating direction algorithm for symmetric nonnegative matrix factorization. The convergence of the algorithm is also proved in detail and strictly. Then we present a modified overlapping community detection method which is based on the presented symmetric nonnegative matrix factorization algorithm. We apply the modified community detection method to several real world networks. The obtained results show the capability of our method in detecting overlapping communities, hubs and outliers. We find that our experimental results have better quality than several competing methods for identifying communities.

Augmented Lagrangian alternating direction method for low-rank minimization via non-convex approximation

This paper concerns the low-rank minimization problems which consist of finding a matrix of minimum rank subject to linear constraints. Many existing approaches, which used the nuclear norm as a convex surrogate of the rank function, usually result in a suboptimal solution. To seek a tighter rank approximation, we develop a non-convex surrogate to approximate the rank function based on the Laplace function. An iterative algorithm based on the augmented Lagrangian multipliers method is developed. Empirical studies for practical applications including robust principal component analysis and low-rank representation demonstrate that our proposed algorithm outperforms many other state-of-the-art convex and non-convex methods developed recently in the literature.

On time delay estimation from a sparse linear prediction perspective.

This paper proposes a sparse linear prediction based algorithm to estimate time difference of arrival. This algorithm unifies the cross correlation method without prewhitening and that with prewhitening via an ℓ2/ℓ1 optimization process, which is solved by an augmented Lagrangian alternating direction method. It also forms a set of time delay estimators that make a tradeoff between prewhitening and non-prewhitening through adjusting a regularization parameter. The effectiveness of the proposed algorithm is demonstrated in noisy and reverberant environments.

Higher-order principal component pursuit via tensor approximation and convex optimization

Recovering the low-rank structure of data matrix from sparse errors arises in the principal component pursuit (PCP). This paper exploits the higher-order generalization of matrix recovery, named higher-order principal component pursuit (HOPCP), since it is critical in multi-way data analysis. Unlike the convexification (nuclear norm) for matrix rank function, the tensorial nuclear norm is still an open problem. While existing preliminary works on the tensor completion field provide a viable way to indicate the low complexity estimate of tensor, therefore, the paper focuses on the low multi-linear rank tensor and adopt its convex relaxation to formulate the convex optimization model of HOPCP. The paper further propose two algorithms for HOPCP based on alternative minimization scheme: the augmented Lagrangian alternating direction method (ALADM) and its truncated higher-order singular value decomposition (ALADM-THOSVD) version. The former can obtain a high accuracy solution while the latter is more efficient to handle the computationally intractable problems. Experimental results on both synthetic data and real magnetic resonance imaging data show the applicability of our algorithms in high-dimensional tensor data processing.

Small infrared target detection based on harmonic and sparse matrix decomposition

Background suppressing is the main technology for infrared target detection. We present a new small infrared target detection (SIRTD) method that is also based on background suppressing. First, a new matrix decomposition model, named harmonic and sparse matrix decomposition (HSMD), is put forward for decomposing an image into a harmonic and a sparse component, which are seen as a background component and a small target component, respectively. Then, an algorithm based on augmented Lagrangian alternating direction method (ALADM) for solving HSMD is described. The main computational cost of the proposed algorithm in each iteration is that of a fast Fourier transform (FFT), which makes the proposed algorithm very fast. By searching for the maximum local energy regions in the target component, the infrared targets can be easily and accurately located. Experimental results on some infrared images show that HSMD solved by ALADM is very suitable for real-time infrared image decomposing and SIRTD.

Nonnegative tensor factorizations using an alternating direction method

The nonnegative tensor (matrix) factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.

Alternating direction methods for classical and ptychographic phase retrieval

In this paper, we show how the augmented Lagrangian alternating direction method (ADM) can be used to solve both the classical and ptychographic phase retrieval problems. We point out the connection between ADM and projection algorithms such as the hybrid input–output algorithm, and compare its performance against standard algorithms for phase retrieval on a number of test images. Our computational experiments show that ADM appears to be less sensitive to the choice of relaxation parameters, and it usually outperforms the existing techniques for both the classical and ptychographic phase retrieval problems.

Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization

The matrix separation problem aims to separate a low-rank matrix and a sparse matrix from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications. Nuclear-norm minimization models have been proposed for matrix separation and proved to yield exact separations under suitable conditions. These models, however, typically require the calculation of a full or partial singular value decomposition at every iteration that can become increasingly costly as matrix dimensions and rank grow. To improve scalability, in this paper, we propose and investigate an alternative approach based on solving a non-convex, low-rank factorization model by an augmented Lagrangian alternating direction method. Numerical studies indicate that the effectiveness of the proposed model is limited to problems where the sparse matrix does not dominate the low-rank one in magnitude, though this limitation can be alleviated by certain data pre-processing techniques. On the other hand, extensive numerical results show that, within its applicability range, the proposed method in general has a much faster solution speed than nuclear-norm minimization algorithms and often provides better recoverability.

Surveillance video processing using compressive sensing

A compressive sensing method combined with decomposition of a matrix formed with image frames of a surveillance video into low rank and sparse matrices is proposed to segment the background and extract moving objects in a surveillance video. The video is acquired by compressive measurements, and the measurements are used to reconstruct the video by a low rank and sparse decomposition of matrix. The low rank component represents the background, and the sparse component is used to identify moving objects in the surveillance video. The decomposition is performed by an augmented Lagrangian alternating direction method. Experiments are carried out to demonstrate that moving objects can be reliably extracted with a small amount of measurements.