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difference of convex algorithms for bilevel programs with applications in hyperparameter selection

MATHEMATICAL PROGRAMMING 2023年第2期198卷 1583-1616页

作者： Ye, Jane J. Yuan, Xiaoming Zeng, Shangzhi Zhang, Jin Univ Victoria Dept Math & Stat Victoria BC Canada Univ Hong Kong Dept Math Hong Kong Peoples R China Southern Univ Sci & Technol SUSTech Int Ctr Math Natl Ctr Appl Math Shenzhen Peng Cheng LabDept Math Shenzhen Peoples R China

In this paper, we present difference of convex algorithms for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel program. We propose two algorithms for solving the reformulated difference of convex program and show their convergence to stationary points under very mild assumptions. Finally we conduct numerical experiments to a bilevel model of support vector machine classification.

关键词： Bilevel program difference of convex algorithm Hyperparameter selection Bilevel model of support vector machine classification

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A Reliable Iteration algorithm for One-Bit Compressive Sensing on the Unit Sphere

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Acta Mathematicae Applicatae Sinica 2024年第3期40卷 801-822页

作者： Yan-cheng LU Ning BI An-hua WAN School of Mathematics Sun Yat-sen UniversityGuangzhou 510275China

The one-bit compressed sensing problem is of fundamental importance in many areas,such as wireless communication,statistics,and so ***,the optimization of one-bit problem coustrained on the unit sphere lacks an algorithm with rigorous mathematical proof of convergence and *** this paper,an iteration algorithm is established based on difference-of-convex algorithm for the one-bit compressed sensing problem constrained on the unit sphere,with iterating formula■,where C is the convex cone generated by the one-bit measurements andη_(1)>η_(2)>1/*** new algorithm is proved to converge as long as the initial point is on the unit sphere and accords with the measurements,and the convergence to the global minimum point of the l_(1)norm is discussed.

关键词： one-bit compressed sensing difference of convex algorithm iteration algorithm ■1-minimization

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Classification of Motor Imagery Task by Using Novel Ensemble Pruning Approach

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IEEE TRANSACTIONS ON FUZZY SYSTEMS 2020年第1期28卷 85-91页

作者： Ali, Muhammad Ammar Ucuncu, Duygu Atas, Pinar Karadayi Ozogur-Akyuz, Sureyya Bahcesehir Univ Fac Engn & Nat Sci Dept Comp Engn TR-34353 Istanbul Turkey Bahcesehir Univ Fac Engn & Nat Sci Dept Math TR-34353 Istanbul Turkey

Brain-computer interface (BCI) assists communication for the disabled and handicapped. It is usually electroencephalogram (EEG) based and uses motor imagery (MI) in its operation. EEG signals are known for being nonstationary and are sensitive to artifacts from various sources such as the physical and mental state of the patient, their mood, their posture, and any external noise or distractions, etc. Processing of this data directly affects the classification accuracy, making it a critical step in any BCI system. Ensemble learning has been used for many kinds of BCI classification applications including MI and P300 event related potential, which has been proven to be robust. The purpose of this paper is to generate an algorithm that uses ensemble pruning method for EEG classification evoked by an MI task. In order to achieve this, we extracted the features of an EEG dataset and trained a range of support vector machines to make a diverse ensemble of classifiers. This ensemble is then pruned by using a novel optimization model by a difference of convex algorithm, which has not been used on EEG data before.

关键词： Electroencephalography Feature extraction Brain modeling Discrete wavelet transforms Task analysis Optimization difference of convex algorithm ensemble pruning motor imagery optimization support vector machines (SVM)

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Fast two-stage segmentation model for images with intensity inhomogeneity

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VISUAL COMPUTER 2020年第6期36卷 1189-1202页

作者： Song, Yangyang Peng, Guohua Northwestern Polytech Univ Dept Appl Math Xian 710072 Shaanxi Peoples R China

Based on the local correntropy-based K-means clustering active contour model, this paper proposes a fast two-stage segmentation method for intensity inhomogeneous images. Under our framework, the segmentation process is split into two stages. In the first stage, we preliminary segment the down-sampled images by the proposed relaxed anisotropic-isotropic local correntropy-based K-means clustering (AILCK) model, which can obtain a coarse segmentation result quickly. Subsequently, in the second stage, we further segment original images by an improved AILCK model, where we use the up-sampled coarse contour obtained by the first stage as the initialization. Following it, to obtain the global minima of energy functions fast, we incorporate a weighted difference of anisotropic and isotropic total variations into relaxed formulation of the two-stage active contour models. And then, we minimize them utilizing the difference-of-convex algorithm and the primal-dual hybrid gradient method. The experimental results on synthetic and real-world images demonstrate that the proposed method can achieve accurate segmentation results for intensity inhomogeneous images in a fast way.

关键词： Image segmentation Two-stage difference of convex algorithm Primal-dual hybrid gradient method Active contour models

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A Weighted difference of Anisotropic and Isotropic Total Variation Model for Image Processing

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SIAM JOURNAL ON IMAGING SCIENCES 2015年第3期8卷 1798-1823页

作者： Lou, Yifei Zeng, Tieyong Osher, Stanley Xin, Jack Univ Texas Dallas Dept Math Sci Richardson TX 75080 USA Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China Univ Calif Los Angeles Dept Math Los Angeles CA 90095 USA Univ Calif Irvine Dept Math Irvine CA 92697 USA

We propose a weighted difference of anisotropic and isotropic total variation (TV) as a regularization for image processing tasks, based on the well-known TV model and natural image statistics. Due to the form of our model, it is natural to compute via a difference of convex algorithm (DCA). We draw its connection to the Bregman iteration for convex problems and prove that the iteration generated from our algorithm converges to a stationary point with the objective function values decreasing monotonically. A stopping strategy based on the stable oscillatory pattern of the iteration error from the ground truth is introduced. In numerical experiments on image denoising, image deblurring, and magnetic resonance imaging (MRI) reconstruction, our method improves on the classical TV model consistently and is on par with representative state-of-the-art methods.

关键词： anisotropic TV isotropic TV weighted difference difference of convex algorithm convergence to stationary points stable oscillatory errors Bregman and split Bregman iterations

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Image Segmentation via Fischer-Burmeister Total Variation and Thresholding

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Advances in Applied Mathematics and Mechanics 2022年第4期14卷 960-988页

作者： TingtingWu Yichen Zhao Zhihui Mao Li Shi Zhi Li Yonghua Zeng School of Science Nanjing University of Posts and TelecommunicationsNanjingJiangsu 210023China The Department of Computer Science and Technology Shanghai Key Laboratory of Multidimensional Information ProcessingEast China Normal UniversityShanghai 200241China College of Field Engineering PLA Army Engineering UniversityNanjingJianhsu 210007China

Image segmentation is a significant problem in image *** this paper,we propose a new two-stage scheme for segmentation based on the Fischer-Burmeister total variation(FBTV).The first stage of our method is to calculate a smooth solution from the FBTV Mumford-Shah ***,we design a new difference of convex algorithm(DCA)with the semi-proximal alternating direction method of multipliers(sPADMM)*** the second stage,we make use of the smooth solution and the K-means method to obtain the segmentation *** simulate images more accurately,a useful operator is introduced,which enables the proposed model to segment not only the noisy or blurry images but the images with missing pixels *** demonstrate the proposed method produces more preferable results comparing with some state-of-the-art methods,especially on the images with missing pixels.

关键词： Image segmentation Fischer-Burmeister total variation difference of convex algorithm sPADMM K-means method.

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A framework for optimization under ambiguity

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ANNALS OF OPERATIONS RESEARCH 2012年第1期193卷 21-47页

作者： Wozabal, David Univ Vienna Dept Business Adm A-1210 Vienna Austria

In this paper, single stage stochastic programs with ambiguous distributions for the involved random variables are considered. Though the true distribution is unknown, existence of a reference measure (P) over cap enables the construction of non-parametric ambiguity sets as Kantorovich balls around (P) over cap. The original stochastic optimization problems are robustified by a worst case approach with respect to these ambiguity sets. The resulting problems are infinite optimization problems and can therefore not be solved computationally by straightforward methods. To nevertheless solve the robustified problems numerically, equivalent formulations as finite dimensional non-convex, semi definite saddle point problems are proposed. Finally an application from portfolio selection is studied for which methods to solve the robust counterpart problems explicitly are proposed and numerical results for sample problems are computed.

关键词： Robust optimization Portfolio management difference of convex algorithm Semi definite programming Expected shortfall Non-convex optimization

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Hybrid Data Fidelity, Weighted difference of TV Norms, Model and algorithm for Noisy Image Deblurring 2

Hybrid Data Fidelity, Weighted Difference of TV Norms, Model...

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2nd International Conference on Robotics and Automation Engineering (ICRAE)

作者： Yu, Yibin Zhang, Jialin Wuyi Univ Sch Informat Engn Jiangmen 529020 Peoples R China

ISBN: (纸本)9781538613061

Although difference of convex model has attracted many research efforts due to its superior performance for image processing, no attention has focused on robust data fidelity for this model. In this paper, we propose a novel model, which combines the 11 and 12 fidelity terms with a weighted difference of anisotropic and isotropic total variation (TV). Since our model takes a new difference form of convex terms, we employ difference of convex algorithm (DCA). In this paper, we adopt the augmented Lagrangian method (ALM) to solve each DCA subproblem of the proposed model, which is named as DCAALM. Experimental results on image deblurring demonstrate that the proposed methods outperform other competing methods in terms of quantitative criteria and perceptual quality.

关键词： image deblurring difference of convex algorithm augmented Lagrangian method hybrid data fidelity

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Topological Interference Alignment via Generalized Low-Rank Optimization with Sequential convex Approximations 19

Topological Interference Alignment via Generalized Low-Rank ...

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IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

作者： Zhang, Fan Wu, Qiong Wang, Hao Shi, Yuanming ShanghaiTech Univ Sch Informat Sci & Technol Shanghai 201210 Peoples R China

ISBN: (纸本)9781538635124

In this paper, we consider solving the topological interference management (TIM) problem by using a generalized low-rank matrix completion (LRMC) model, thereby maximizing the achievable degrees of freedom (DoF) only based on the network connectivity information. The LRMC problem is NP-hard due to the nonconvex rank objective function. The nuclear norm relaxation fails as it always returns a full-rank matrix in our model. Another approach named Riemannian Pursuit (RP) is often inefficient for finding highly accurate feasible solutions. We thus propose a novel Generalized Low-Rank Optimization along with the difference of convex algorithm (GLRO-DCA), which aims to find a low-rank solution while always keeping the feasiblity. The GLRO-DCA increases the rank consecutively and solves the associated fixed-rank LRMC problem, where the generalized fixed-rank LRMC problem is reformulated by minimizing the difference between the nuclear norm and the Ky Fan norm and solved by the DCA. We accelerate the DCA by applying extrapolation techniques to improve the computational efficiency. Numerical results exhibit the ability of our proposed GLRO-DCA for the TIM problem to find low-rank solutions, which is superior to the existing nuclear norm relaxation approach and the RP approach.

关键词： Interference alignment degrees of freedom low-rank matrix completion Ky Fan norm difference of convex algorithm

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PIECEWISE-AFFINE CLASSIFIERS IN SUPPORT VECTOR MACHINES

PIECEWISE-AFFINE CLASSIFIERS IN SUPPORT VECTOR MACHINES

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作者： Matthew T. Miller NAVAL POSTGRADUATE SCHOOL

学位级别：硕士

The Support Vector Machine (SVM) model has been a topic of study for over twenty years, and novel approaches to the classification problem using SVM continue to be established. In this work, we develop a new, nonlinear version of SVM based on a piecewise-affine classifier. This class of classifiers constitutes a tractable class beyond the affine functions that enables approximation of nonparametric SVM in high dimensions. We solve the resulting Piecewise-Affine SVM (PA-SVM) model using the difference-of-convex algorithm (DCA) and a stochastic gradient descent (SGD) algorithm. The PA-SVM model is nonconvex, and the algorithms generally only provide locally optimal solutions. Still, they provide for a robust, capable classifier. Results show that by using DCA, the PA-SVM model can significantly reduce training misclassifications relative to the common Affine SVM (A-SVM) model by as much as 92%. Additionally, we show that test set errors can be reduced by as much as 67% compared to A-SVM. We find that solutions are more affected by the number of pieces employed rather than by regularization penalties. These results come from applying the PA-SVM model to three real-world data sets whose total features range from 16 to 41 and whose total observations range from 194 to 1,553.

关键词： optimization classification piecewise-affine difference of convex difference of convex algorithm support vector machine stochastic gradient descent nonconvex nonlinear

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