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A primal perspective for indefinite kernel SVM problem

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Frontiers of Computer Science 2020年第2期14卷 349-363页

作者： Hui XUE Haiming XU Xiaohong CHEN Yunyun WANG School of Computer Science and Engineering Southeast UniversityNanjing 210096China Key Laboratory of Computer Network and Information Integration(Southeast University) MOENanjing 210096China College of Science Nanjing University of Aeronautics and AstronauticsNanjing 210016China School of Computer Science Nanjing University of Posts and TelecommunicationsNanjing 210046China

indefinite kernel support vector machine(IKSVM)has recently attracted increasing attentions in machine learning.Since IKSVM essentially is a non-convex problem,existing algorithms either change the spectrum of indefinite kernel directly but risking losing some valuable information or solve the dual form of IKSVM whereas suffering from a dual gap problem.In this paper,we propose a primal perspective for solving the problem.That is,we directly focus on the primal form of IKSVM and present a novel algorithm termed as IKSVM-DC for binary and multi-class classification.Concretely,according to the characteristics of the spectrum for the indefinite kernel matrix,IKSVM-DC decomposes the primal function into the subtraction of two convex functions as a difference of convex functions(DC)programming.To accelerate convergence rate,IKSVM-DC combines the classical DC algorithm with a line search step along the descent direction at each iteration.Furthermore,we construct a multi-class IKSVM model which can classify multiple classes in a unified form.A theoretical analysis is then presented to validate that IKSVM-DC can converge to a local minimum.Finally,we conduct experiments on both binary and multi-class datasets and the experimental results show that IKSVM-DC is superior to other state-of-the-art IKSVM algorithms.

关键词： indefinite kernel support vector machine multi-class classification non-convex optimization

A maximum margin clustering algorithm based on indefinite kernels

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Frontiers of Computer Science 2019年第4期13卷 813-827页

作者： Hui XUE Sen LI Xiaohong CHEN Yunyun WANG School of Computer Science and Engineering Southeast University Nanjing 210096 Chin Key Laboratory of Computer Network and Information Integration (Southeast University) Ministry of Education Nanjing 210096 China College of Science Nanjing University of Aeronautics and Astronautics Nanjing 210016 China Department of Computer Science and Engineering Nanjing University of Posts and Telecommunications Nanjing 210046 China

indefinite kernels have attracted more and more attentions in machine learning due to its wider application scope than usual positive definite kernels. However, the research about indefinite kernel clustering is relatively scarce. Furthermore, existing clustering methods are mainly designed based on positive definite kernels which are incapable in indefinite kernel scenarios. In this paper, we propose a novel indefinite kernel clustering algorithm termed as indefinite kernel maximum margin clustering (IKMMC) based on the state-of-the-art maximum margin clustering (MMC) model. IKMMC tries to find a proxy positive definite kernel to approximate the original indefinite one and thus embeds a new F-norm regularizer in the objective function to measure the diversity of the two kernels, which can be further optimized by an iterative approach. Concretely, at each iteration, given a set of initial class labels, IKMMC firstly transforms the clustering problem into a classification one solved by indefinite kernel support vector machine (IKSVM) with an extra class balance constraint and then the obtained prediction labels will be used as the new input class labels at next iteration until the error rate of prediction is smaller than a prespecified tolerance. Finally, IKMMC utilizes the prediction labels at the last iteration as the expected indices of clusters.Moreover, we further extend IKMMC from binary clustering problems to more complex multi-class scenarios. Experimental results have shown the superiority of our algorithms.

关键词： indefinite kernel maximum margin clustering support vector machine kernel method

Cyclic Group Action of Composite Order on indefinite 4-manifolds

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Communications in Mathematical Research 2011年第3期27卷 200-206页

作者： LI HONG-XIA LIU XI-MIN Department of Applied Mathematics Shanghai Maritime University Shanghai 201306 Department of Mathematics South China University of Technology Guangzhou 510641

In this paper, we study the possibilities for several kinds of topological, locally linear cyclic group actions of non-prime order on some closed, simply connected 4-manifolds with indefinite intersection form. Especi... 详细信息

关键词： periodic map 4-manifold indefinite permutation representation

Negative Refraction in indefinite Materials

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Negative Refraction in Indefinite Materials

Approximation algorithms for indefinite complex quadratic maximization problems

2012 China Functional Materials Technology and Industry Forum

作者： Hong-ya Wu Ji Zhou State key laboratory of new ceramic and fine processing Department of materials science and engineeringTsinghua University

Negative refraction is a fantastic optical property and attracts more and more attentions. As first proposed,materials with negative permittivity and negative permeability simultaneously exhibit negative refraction.Recently,negative refraction was found in the indefinite medium based on the strong anisotropy of permittivity.In this paper,the theory of the negative refraction in the anisotropic materials and the research progress in recent years were reviewed.

关键词： Anisotropic Negative refraction indefinite

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Science China Mathematics 2010年第10期53卷 2697-2708页

作者： HUANG Yongwei1,& ZHANG Shuzhong2 1Department of Electronic and Computer Engineering,The Hong Kong University of Science and Technology,Kowloon,Hong Kong,China 2Department of Systems Engineering and Engineering Management,The Chinese University of Hong Kong,Shatin,Hong Kong,China Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Kowloon Hong Kong China Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin Hong Kong China

In this paper,we consider the following indefinite complex quadratic maximization problem: maximize zHQz,subject to zk ∈ C and zkm = 1,k = 1,...,n,where Q is a Hermitian matrix with trQ = 0,z ∈ Cn is the decision vector,and m 3.An (1/log n) approximation algorithm is presented for such problem.Furthermore,we consider the above problem where the objective matrix Q is in bilinear form,in which case a 0.7118 cos mπ 2approximation algorithm can be constructed.In the context of quadratic optimization,various extensions and connections of the model are discussed.

关键词： indefinite Hermitian matrix randomized algorithms approximation ratio semidefinite programming relaxation

Spectrum of a Class of Difference Operators with indefinite Weights

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Journal of Applied Mathematics and Physics 2020年第4期8卷 727-736页

作者： Congmin Yang Yunlan Gao Kang Sun College of Sciences Inner Mongolia University of Technology Hohhot China

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

关键词： Spectrum Difference Operator Coupled Boundary Conditions indefinite Weight

Clinical significance and management of Barrett's esophagus with epithelial changes indefinite for dysplasia

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World Journal of Gastrointestinal Pharmacology and Therapeutics 2016年第3期7卷 406-411页

作者： Prashanthi N Thota Gaurav Kistangari Ashwini K Esnakula David Hernandez Gonzalo Xiu-Li Liu Department of Gastroenterology Digestive Disease Institute Cleveland Clinic Department of Pathology Immunology and Laboratory Medicine University of Florida College of Medicine

Barrett's esophagus(BE) is defined as the extension of salmon-colored mucosa into the tubular esophagus ≥1 cm proximal to the gastroesophageal junction with biopsy confirmation of intestinal metaplasia. Patients with BE are at increased risk of esophageal adenocarcinoma(EAC), and undergo endoscopic surveillance biopsies to detect dysplasia or early EAC. Dysplasia in BE is classified as no dysplasia, indefinite for dysplasia(IND), low grade dysplasia(LGD) or high grade dysplasia(HGD). Biopsies are diagnosed as IND when the epithelial abnor-malities are not sufficient to diagnose dysplasia or the nature of the epithelial abnormalities is uncertain due to inflammation or technical issues. Specific diagnostic criteria for IND are not well established and its clinical significance and management has not been well studied. Previous studies have focused on HGD in BE and led to changes and improvement in the management of BE with HGD and early EAC. Only recently, IND and LGD in BE have become focus of intense study. This review summarizes the definition, neoplastic risk and clinical management of BE IND.

关键词： Barrett’s esophagus Dysplasia Progression Biomarkers Esophageal adenocarcinoma indefinite for Dysplasia

Solutions and solvabilities of general binary quadric indefinite equations

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Chinese Science Bulletin 1997年第7期42卷 541-545页

作者： LiU LiAnhui Traditional Chinese Medical College, Hefei 230038, China Anhui Traditional Chinese Medical College Hefei China

To this day for special m and small |c|, solvable, c are obtained about general equations x^2-my^2=c with positive squarefree integer m and integer c by Ankeny, Mollin et al. Afterwards, Zhang in ref. [1] greatly developed their correspondent results and gave all proper integer solutions and solvable c about the equations for general m. Then Lu in ref. [2] gave results for small |q| about general binary quadric indefinite

关键词： general binary quadric indefinite equation proper integer solution solvability.

PRECONDITIONING MIXED ELEMENT METHOD FOR NONSELFADJOINT AND indefinite SECOND ORDER ELLIPTIC PROBLEMS

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1999年第1期8卷 67-82页

作者：陈金如陈文斌

In this paper, an equivalence between mixed element method and nonconforming element method for nonselfad joint and indefinite second order elliptic problems is established without using any bubble functions. It is proved that the H1-condition number of preconditioned operator Bh-1Ah is uniformly bounded and its Bh-singular values cluster in a positive finite interval, where Ah is the equivalent nonconforming element discretization of nonselfad joint and indefinite second order elliptic operator A, Bh is usual noncon forming element discretization of selfadjoint and positive definite second order elliptic operator B. Finally a simple V-cycle multigrid implementation of Bh-1 is given.

关键词： Preconditioning mixed element indefinite elliptic problems.