scholarly journals Efficient Proximal Gradient Algorithms for Joint Graphical Lasso

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1623
Author(s):  
Jie Chen ◽  
Ryosuke Shimmura ◽  
Joe Suzuki
Keyword(s):  
Graphical Model ◽  
State Of The Art ◽  
High Accuracy ◽  
Alternating Direction Method ◽  
Main Approach ◽  
First Order ◽  
Gradient Algorithms ◽  
Graphical Lasso ◽  
Alternating Direction ◽  
Accuracy And Precision

We consider learning as an undirected graphical model from sparse data. While several efficient algorithms have been proposed for graphical lasso (GL), the alternating direction method of multipliers (ADMM) is the main approach taken concerning joint graphical lasso (JGL). We propose proximal gradient procedures with and without a backtracking option for the JGL. These procedures are first-order methods and relatively simple, and the subproblems are solved efficiently in closed form. We further show the boundedness for the solution of the JGL problem and the iterates in the algorithms. The numerical results indicate that the proposed algorithms can achieve high accuracy and precision, and their efficiency is competitive with state-of-the-art algorithms.

Influence of Thermal Radiation on the Temperature Distribution in a Semi-Transparent Solid

1979 ◽  
Vol 101 (1) ◽  
pp. 76-80 ◽  
Author(s):  
D. W. Amlin ◽  
S. A. Korpela
Keyword(s):  
Temperature Distribution ◽  
Nonlinear Partial Differential Equations ◽  
Alternating Direction Method ◽  
Temperature Profiles ◽  
Rectangular Region ◽  
Differential Approximation ◽  
First Order ◽  
Alternating Direction ◽  
Plane Slab ◽  
Good Agreement

The importance of radiation on the temperature distribution in a semi-transparent solid is reported. The first-order differential approximation of radiation is combined with conduction analysis to investigate the temperature profiles in a plane slab and a rectangular region. The coupled nonlinear partial differential equations are solved numerically by either a standard implicit or an implicit alternating direction method. Results obtained for opaque boundaries are in good agreement with exact formulations found in the literature. An extension to partially transparent boundaries is made and results presented.

scholarly journals Two Efficient Algorithms for Orthogonal Nonnegative Matrix Factorization

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jing Wu ◽  
Bin Chen ◽  
Tao Han
Keyword(s):  
Gene Expression ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
State Of The Art ◽  
Nonnegative Matrix ◽  
Alternating Direction Method ◽  
Data Matrix ◽  
Popular Method ◽  
Alternating Direction ◽  
Expression Classification

Nonnegative matrix factorization (NMF) is a popular method for the multivariate analysis of nonnegative data. It involves decomposing a data matrix into a product of two factor matrices with all entries restricted to being nonnegative. Orthogonal nonnegative matrix factorization (ONMF) has been introduced recently. This method has demonstrated remarkable performance in clustering tasks, such as gene expression classification. In this study, we introduce two convergence methods for solving ONMF. First, we design a convergent orthogonal algorithm based on the Lagrange multiplier method. Second, we propose an approach that is based on the alternating direction method. Finally, we demonstrate that the two proposed approaches tend to deliver higher-quality solutions and perform better in clustering tasks compared with a state-of-the-art ONMF.

scholarly journals Fusing Multiple Multiband Images

2018 ◽  
Vol 4 (10) ◽  
pp. 118 ◽  
Author(s):  
Reza Arablouei
Keyword(s):  
State Of The Art ◽  
Likelihood Estimation ◽  
Alternating Direction Method ◽  
Natural Images ◽  
Hyperspectral Images ◽  
Superior Performance ◽  
Method Of Multipliers ◽  
Alternating Direction ◽  
Abrupt Changes ◽  
Fused Image

High-resolution hyperspectral images are in great demand but hard to acquire due to several existing fundamental and technical limitations. A practical way around this is to fuse multiple multiband images of the same scene with complementary spatial and spectral resolutions. We propose an algorithm for fusing an arbitrary number of coregistered multiband, i.e., panchromatic, multispectral, or hyperspectral, images through estimating the endmember and their abundances in the fused image. To this end, we use the forward observation and linear mixture models and formulate an appropriate maximum-likelihood estimation problem. Then, we regularize the problem via a vector total-variation penalty and the non-negativity/sum-to-one constraints on the endmember abundances and solve it using the alternating direction method of multipliers. The regularization facilitates exploiting the prior knowledge that natural images are mostly composed of piecewise smooth regions with limited abrupt changes, i.e., edges, as well as coping with potential ill-posedness of the fusion problem. Experiments with multiband images constructed from real-world hyperspectral images reveal the superior performance of the proposed algorithm in comparison with the state-of-the-art algorithms, which need to be used in tandem to fuse more than two multiband images.

scholarly journals An Alternating Direction Method for Mixed Gaussian Plus Impulse Noise Removal

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Si Wang ◽  
Ting-Zhu Huang ◽  
Xi-le Zhao ◽  
Jun Liu
Keyword(s):  
Total Variation ◽  
Numerical Experiments ◽  
Impulse Noise ◽  
State Of The Art ◽  
Noise Removal ◽  
Alternating Direction Method ◽  
Impulse Noise Removal ◽  
Alternating Direction ◽  
Total Variation Model ◽  
Blurred Images

A combined total variation and high-order total variation model is proposed to restore blurred images corrupted by impulse noise or mixed Gaussian plus impulse noise. We attack the proposed scheme with an alternating direction method of multipliers (ADMM). Numerical experiments demonstrate the efficiency of the proposed method and the performance of the proposed method is competitive with the existing state-of-the-art methods.

scholarly journals Alternating Direction Methods for Latent Variable Gaussian Graphical Model Selection

2013 ◽  
Vol 25 (8) ◽  
pp. 2172-2198 ◽  
Author(s):  
Shiqian Ma ◽  
Lingzhou Xue ◽  
Hui Zou
Keyword(s):  
Model Selection ◽  
Convex Optimization ◽  
Optimization Problem ◽  
Graphical Model ◽  
Sparse Matrix ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Convex Optimization Problem ◽  
Alternating Direction Methods ◽  
Alternating Direction

Chandrasekaran, Parrilo, and Willsky ( 2012 ) proposed a convex optimization problem for graphical model selection in the presence of unobserved variables. This convex optimization problem aims to estimate an inverse covariance matrix that can be decomposed into a sparse matrix minus a low-rank matrix from sample data. Solving this convex optimization problem is very challenging, especially for large problems. In this letter, we propose two alternating direction methods for solving this problem. The first method is to apply the classic alternating direction method of multipliers to solve the problem as a consensus problem. The second method is a proximal gradient-based alternating-direction method of multipliers. Our methods take advantage of the special structure of the problem and thus can solve large problems very efficiently. A global convergence result is established for the proposed methods. Numerical results on both synthetic data and gene expression data show that our methods usually solve problems with 1 million variables in 1 to 2 minutes and are usually 5 to 35 times faster than a state-of-the-art Newton-CG proximal point algorithm.

scholarly journals MADAM: a parallel exact solver for max-cut based on semidefinite programming and ADMM

2021 ◽  
Vol 80 (2) ◽  
pp. 347-375
Author(s):  
Timotej Hrga ◽  
Janez Povh
Keyword(s):  
State Of The Art ◽  
Inequality Constraints ◽  
Alternating Direction Method ◽  
Dual Variable ◽  
Maximum Weight ◽  
New Approach ◽  
Parallel Solver ◽  
Alternating Direction ◽  
Computationally Expensive ◽  
Serial Algorithm

AbstractWe present , a parallel semidefinite-based exact solver for Max-Cut, a problem of finding the cut with the maximum weight in a given graph. The algorithm uses the branch and bound paradigm that applies the alternating direction method of multipliers as the bounding routine to solve the basic semidefinite relaxation strengthened by a subset of hypermetric inequalities. The benefit of the new approach is a less computationally expensive update rule for the dual variable with respect to the inequality constraints. We provide a theoretical convergence of the algorithm as well as extensive computational experiments with this method, to show that our algorithm outperforms state-of-the-art approaches. Furthermore, by combining algorithmic ingredients from the serial algorithm, we develop an efficient distributed parallel solver based on MPI.

scholarly journals Adaptive Image Restoration via a Relaxed Regularization of Mean Curvature

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mingxi Ma ◽  
Jun Zhang ◽  
Chengzhi Deng ◽  
Zhaoyang Liu ◽  
Yuanyun Wang
Keyword(s):  
Image Restoration ◽  
Mean Curvature ◽  
Complexity Analysis ◽  
State Of The Art ◽  
Alternating Direction Method ◽  
Natural Images ◽  
Convergence Theory ◽  
Relaxation Model ◽  
Alternating Direction ◽  
Effectiveness And Efficiency

In this paper, a new relaxation model based on mean curvature for adaptive image restoration is proposed. To solve the problem efficiently, an alternating direction method of multipliers (ADMMs) is proposed. Furthermore, a rigorous convergence theory of the proposed algorithm is established. We also give the complexity analysis of our proposed method. Experimental results are provided to demonstrate the effectiveness and efficiency of the proposed method over a state-of-the-art method on synthetic and natural images.

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