一种求解分类问题的新算法

本文提出了基于支持向量回归机(SVR)的一种新分类算法.它和标准的支持向量机(SVM)不同:标准的支持向量机(SVM)采用固定的模度量间隔且最优化问题与参数有关.本文中我们可以用任意模度量间隔,得到的最优化问题是无参数的线性规划问题,避免了参数选择.数值试验表明了该算法的有效性.     

Two-Dimensional Solution Surface for Weighted Support Vector Machines

The support vector machine (SVM) is a popular learning method for binary classification. Standard SVMs treat all the data points equally, but in some practical problems it is more natural to assign different weights to observations from different classes. This leads to a broader class of learning, the so-called weighted SVMs (WSVMs), and one of their important applications is to estimate class probabilities besides learning the classification boundary. There are two parameters associated with the WSVM optimization problem: one is the regularization parameter and the other is the weight parameter. In this article, we first establish that the WSVM solutions are jointly piecewise-linear with respect to both the regularization and weight parameter. We then develop a state-of-the-art algorithm that can compute the entire trajectory of the WSVM solutions for every pair of the regularization parameter and the weight parameter at a feasible computational cost. The derived two-dimensional solution surface provides theoretical insight on the behavior of the WSVM solutions. Numerically, the algorithm can greatly facilitate the implementation of the WSVM and automate the selection process of the optimal regularization parameter. We illustrate the new algorithm on various examples. This article has online supplementary materials.     

Improved twin support vector machine

We improve the twin support vector machine(TWSVM)to be a novel nonparallel hyperplanes classifier,termed as ITSVM(improved twin support vector machine),for binary classification.By introducing the diferent Lagrangian functions for the primal problems in the TWSVM,we get an improved dual formulation of TWSVM,then the resulted ITSVM algorithm overcomes the common drawbacks in the TWSVMs and inherits the essence of the standard SVMs.Firstly,ITSVM does not need to compute the large inverse matrices before training which is inevitable for the TWSVMs.Secondly,diferent from the TWSVMs,kernel trick can be applied directly to ITSVM for the nonlinear case,therefore nonlinear ITSVM is superior to nonlinear TWSVM theoretically.Thirdly,ITSVM can be solved efciently by the successive overrelaxation(SOR)technique or sequential minimization optimization(SMO)method,which makes it more suitable for large scale problems.We also prove that the standard SVM is the special case of ITSVM.Experimental results show the efciency of our method in both computation time and classification accuracy.     

Robust multicategory support vector machines using difference convex algorithm

The support vector machine (SVM) is one of the most popular classification methods in the machine learning literature. Binary SVM methods have been extensively studied, and have achieved many successes in various disciplines. However, generalization to multicategory SVM (MSVM) methods can be very challenging. Many existing methods estimate k functions for k classes with an explicit sum-to-zero constraint. It was shown recently that such a formulation can be suboptimal. Moreover, many existing MSVMs are not Fisher consistent, or do not take into account the effect of outliers. In this paper, we focus on classification in the angle-based framework, which is free of the explicit sum-to-zero constraint, hence more efficient, and propose two robust MSVM methods using truncated hinge loss functions. We show that our new classifiers can enjoy Fisher consistency, and simultaneously alleviate the impact of outliers to achieve more stable classification performance. To implement our proposed classifiers, we employ the difference convex algorithm for efficient computation. Theoretical and numerical results obtained indicate that for problems with potential outliers, our robust angle-based MSVMs can be very competitive among existing methods.     

从支持向量机到非平行支持向量机

非平行支持向量机是支持向量机的延伸,受到了广泛的关注.非平行支持向量机构造允许非平行的支撑超平面,可以描述不同类别之间的数据分布差异,从而适用于更广泛的问题.然而,对非平行支持向量机模型与支持向量机模型之间的关系研究较少,且尚未有等价于标准支持向量机模型的非平行支持向量机模型.从支持向量机出发,构造出新的非平行支持向量机模型,该模型不仅可以退化为标准支持向量机,保留了支持向量机的稀疏性和核函数可扩展性.同时,可以描述不同类别之间的数据分布差异,适用于更广泛的非平行结构数据等.最后,通过实验初步验证了所提模型的有效性.     

An Efficient Algorithm for Computing the HHSVM and Its Generalizations

The hybrid Huberized support vector machine (HHSVM) has proved its advantages over the ?₁ support vector machine (SVM) in terms of classification and variable selection. Similar to the ?₁ SVM, the HHSVM enjoys a piecewise linear path property and can be computed by a least-angle regression (LARS)-type piecewise linear solution path algorithm. In this article, we propose a generalized coordinate descent (GCD) algorithm for computing the solution path of the HHSVM. The GCD algorithm takes advantage of a majorization–minimization trick to make each coordinatewise update simple and efficient. Extensive numerical experiments show that the GCD algorithm is much faster than the LARS-type path algorithm. We further extend the GCD algorithm to solve a class of elastic net penalized large margin classifiers, demonstrating the generality of the GCD algorithm. We have implemented the GCD algorithm in a publicly available R package gcdnet.     

Application of SVM and ANN for image retrieval

This paper presents a new, scaling and rotation invariant encoding scheme for shapes. Support vector machines (SVMs) and artificial neural networks (ANNs) are used for the classifications of shapes encoded by the new method. The SVM classification accuracy rate is 95.9 ∓ 2.9% in 14 categories and 79.2 ∓ 2.1% in 40 categories. This shows that SVM is one of the best tools for classification problems. The experimental results showed that SVM achieved better performance than ANN. A sensitivity test is performed to show that SVM is quite robust against different parameter values. In addition, our coding method is comparable to previous coding scheme in terms of SVM and ANN performance.     

Reinforced Angle-Based Multicategory Support Vector Machines

The support vector machine (SVM) is a very popular classification tool with many successful applications. It was originally designed for binary problems with desirable theoretical properties. Although there exist various multicategory SVM (MSVM) extensions in the literature, some challenges remain. In particular, most existing MSVMs make use of k classification functions for a k-class problem, and the corresponding optimization problems are typically handled by existing quadratic programming solvers. In this article, we propose a new group of MSVMs, namely, the reinforced angle-based MSVMs (RAMSVMs), using an angle-based prediction rule with k ? 1 functions directly. We prove that RAMSVMs can enjoy Fisher consistency. Moreover, we show that the RAMSVM can be implemented using the very efficient coordinate descent algorithm on its dual problem. Numerical experiments demonstrate that our method is highly competitive in terms of computational speed, as well as classification prediction performance. Supplemental materials for the article are available online.     

基于提升小波的基因芯片数据的分类预测

针对肿瘤的早期诊断,提出了一种基于提升小波变换的特征提取的方法,对肿瘤数据样本进行分析鉴别.该方法利用提升小波变换对190例肝癌（包括对照）和107例肺癌（包括对照）基因表达谱芯片数据进行处理后,提取信号的低频信息,经支持向量机训练学习,构造分类器模型,用于癌和非癌样本的区分甄别.实验结果表明,经提升小波变换提取的特征基因,送入分类器中能得到较高的分类率,且在支持向量机中选取线性核函数或径向基函数都能达到较好的分类效果.通过随机选取的20例基因表达谱芯片样本,对所建立的模型进行了测试,获得了很好的效果,因此,本文提出的方法对肿瘤的诊断有一定的应用意义.     

A comparative study on large scale kernelized support vector machines

Kernelized support vector machines (SVMs) belong to the most widely used classification methods. However, in contrast to linear SVMs, the computation time required to train such a machine becomes a bottleneck when facing large data sets. In order to mitigate this shortcoming of kernel SVMs, many approximate training algorithms were developed. While most of these methods claim to be much faster than the state-of-the-art solver LIBSVM, a thorough comparative study is missing. We aim to fill this gap. We choose several well-known approximate SVM solvers and compare their performance on a number of large benchmark data sets. Our focus is to analyze the trade-off between prediction error and runtime for different learning and accuracy parameter settings. This includes simple subsampling of the data, the poor-man’s approach to handling large scale problems. We employ model-based multi-objective optimization, which allows us to tune the parameters of learning machine and solver over the full range of accuracy/runtime trade-offs. We analyze (differences between) solvers by studying and comparing the Pareto fronts formed by the two objectives classification error and training time. Unsurprisingly, given more runtime most solvers are able to find more accurate solutions, i.e., achieve a higher prediction accuracy. It turns out that LIBSVM with subsampling of the data is a strong baseline. Some solvers systematically outperform others, which allows us to give concrete recommendations of when to use which solver.     

Kernel Logistic Regression and the Import Vector Machine

The support vector machine (SVM) is known for its good performance in two-class classification, but its extension to multiclass classification is still an ongoing research issue. In this article, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only performs as well as the SVM in two-class classification, but also can naturally be generalized to the multiclass case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the support points of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM. This gives the IVM a potential computational advantage over the SVM.     

Support vector machine classifiers with uncertain knowledge sets via robust optimization

In this article we study support vector machine (SVM) classifiers in the face of uncertain knowledge sets and show how data uncertainty in knowledge sets can be treated in SVM classification by employing robust optimization. We present knowledge-based SVM classifiers with uncertain knowledge sets using convex quadratic optimization duality. We show that the knowledge-based SVM, where prior knowledge is in the form of uncertain linear constraints, results in an uncertain convex optimization problem with a set containment constraint. Using a new extension of Farkas' lemma, we reformulate the robust counterpart of the uncertain convex optimization problem in the case of interval uncertainty as a convex quadratic optimization problem. We then reformulate the resulting convex optimization problems as a simple quadratic optimization problem with non-negativity constraints using the Lagrange duality. We obtain the solution of the converted problem by a fixed point iterative algorithm and establish the convergence of the algorithm. We finally present some preliminary results of our computational experiments of the method.     

Comprehensible credit scoring models using rule extraction from support vector machines

In recent years, support vector machines (SVMs) were successfully applied to a wide range of applications. However, since the classifier is described as a complex mathematical function, it is rather incomprehensible for humans. This opacity property prevents them from being used in many real-life applications where both accuracy and comprehensibility are required, such as medical diagnosis and credit risk evaluation. To overcome this limitation, rules can be extracted from the trained SVM that are interpretable by humans and keep as much of the accuracy of the SVM as possible. In this paper, we will provide an overview of the recently proposed rule extraction techniques for SVMs and introduce two others taken from the artificial neural networks domain, being Trepan and G-REX. The described techniques are compared using publicly available datasets, such as Ripley’s synthetic dataset and the multi-class iris dataset. We will also look at medical diagnosis and credit scoring where comprehensibility is a key requirement and even a regulatory recommendation. Our experiments show that the SVM rule extraction techniques lose only a small percentage in performance compared to SVMs and therefore rank at the top of comprehensible classification techniques.     

Multiclass Probability Estimation With Support Vector Machines

Multiclass classification and probability estimation have important applications in data analytics. Support vector machines (SVMs) have shown great success in various real-world problems due to their high classification accuracy. However, one main limitation of standard SVMs is that they do not provide class probability estimates, and thus fail to offer uncertainty measure about class prediction. In this article, we propose a simple yet effective framework to endow kernel SVMs with the feature of multiclass probability estimation. The new probability estimator does not rely on any parametric assumption on the data distribution, therefore, it is flexible and robust. Theoretically, we show that the proposed estimator is asymptotically consistent. Computationally, the new procedure can be conveniently implemented using standard SVM softwares. Our extensive numerical studies demonstrate competitive performance of the new estimator when compared with existing methods such as multiple logistic regression, linear discrimination analysis, tree-based methods, and random forest, under various classification settings. Supplementary materials for this article are available online.     

A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training

Support vector machines (SVMs) training may be posed as a large quadratic program (QP) with bound constraints and a single linear equality constraint. We propose a (block) coordinate gradient descent method for solving this problem and, more generally, linearly constrained smooth optimization. Our method is closely related to decomposition methods currently popular for SVM training. We establish global convergence and, under a local error bound assumption (which is satisfied by the SVM QP), linear rate of convergence for our method when the coordinate block is chosen by a Gauss-Southwell-type rule to ensure sufficient descent. We show that, for the SVM QP with n variables, this rule can be implemented in O(n) operations using Rockafellar’s notion of conformal realization. Thus, for SVM training, our method requires only O(n) operations per iteration and, in contrast to existing decomposition methods, achieves linear convergence without additional assumptions. We report our numerical experience with the method on some large SVM QP arising from two-class data classification. Our experience suggests that the method can be efficient for SVM training with nonlinear kernel.     

MOP/GP models for machine learning

Techniques for machine learning have been extensively studied in recent years as effective tools in data mining. Although there have been several approaches to machine learning, we focus on the mathematical programming (in particular, multi-objective and goal programming; MOP/GP) approaches in this paper. Among them, Support Vector Machine (SVM) is gaining much popularity recently. In pattern classification problems with two class sets, its idea is to find a maximal margin separating hyperplane which gives the greatest separation between the classes in a high dimensional feature space. This task is performed by solving a quadratic programming problem in a traditional formulation, and can be reduced to solving a linear programming in another formulation. However, the idea of maximal margin separation is not quite new: in the 1960s the multi-surface method (MSM) was suggested by Mangasarian. In the 1980s, linear classifiers using goal programming were developed extensively.This paper presents an overview on how effectively MOP/GP techniques can be applied to machine learning such as SVM, and discusses their problems.     

Pattern classification by goal programming and support vector machines

Support Vector Machines (SVMs) are now very popular as a powerful method in pattern classification problems. One of main features of SVMs is to produce a separating hyperplane which maximizes the margin in feature space induced by nonlinear mapping using kernel function. As a result, SVMs can treat not only linear separation but also nonlinear separation. While the soft margin method of SVMs considers only the distance between separating hyperplane and misclassified data, we propose in this paper multi-objective programming formulation considering surplus variables. A similar formulation was extensively researched in linear discriminant analysis mostly in 1980s by using Goal Programming(GP). This paper compares these conventional methods such as SVMs and GP with our proposed formulation through several examples.Received: September 2003, Revised: December 2003,     

Tuning membership functions of kernel fuzzy classifiers by maximizing margins

We propose two methods for tuning membership functions of a kernel fuzzy classifier based on the idea of SVM (support vector machine) training. We assume that in a kernel fuzzy classifier a fuzzy rule is defined for each class in the feature space. In the first method, we tune the slopes of the membership functions at the same time so that the margin between classes is maximized under the constraints that the degree of membership to which a data sample belongs is the maximum among all the classes. This method is similar to a linear all-at-once SVM. We call this AAO tuning. In the second method, we tune the membership function of a class one at a time. Namely, for a class the slope of the associated membership function is tuned so that the margin between the class and the remaining classes is maximized under the constraints that the degrees of membership for the data belonging to the class are large and those for the remaining data are small. This method is similar to a linear one-against-all SVM. This is called OAA tuning. According to the computer experiment for fuzzy classifiers based on kernel discriminant analysis and those with ellipsoidal regions, usually both methods improve classification performance by tuning membership functions and classification performance by AAO tuning is slightly better than that by OAA tuning.     

SVM-Maj: a majorization approach to linear support vector machines with different hinge errors

Support vector machines (SVM) are becoming increasingly popular for the prediction of a binary dependent variable. SVMs perform very well with respect to competing techniques. Often, the solution of an SVM is obtained by switching to the dual. In this paper, we stick to the primal support vector machine problem, study its effective aspects, and propose varieties of convex loss functions such as the standard for SVM with the absolute hinge error as well as the quadratic hinge and the Huber hinge errors. We present an iterative majorization algorithm that minimizes each of the adaptations. In addition, we show that many of the features of an SVM are also obtained by an optimal scaling approach to regression. We illustrate this with an example from the literature and do a comparison of different methods on several empirical data sets.     

Sparse Penalized Forward Selection for Support Vector Classification

We propose a new binary classification and variable selection technique especially designed for high-dimensional predictors. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection along with classification. By adding an ?₁-type penalty to the loss function, common classification methods such as logistic regression or support vector machines (SVM) can perform variable selection. Existing penalized SVM methods all attempt to jointly solve all the parameters involved in the penalization problem altogether. When data dimension is very high, the joint optimization problem is very complex and involves a lot of memory allocation. In this article, we propose a new penalized forward search technique that can reduce high-dimensional optimization problems to one-dimensional optimization by iterating the selection steps. The new algorithm can be regarded as a forward selection version of the penalized SVM and its variants. The advantage of optimizing in one dimension is that the location of the optimum solution can be obtained with intelligent search by exploiting convexity and a piecewise linear or quadratic structure of the criterion function. In each step, the predictor that is most able to predict the outcome is chosen in the model. The search is then repeatedly used in an iterative fashion until convergence occurs. Comparison of our new classification rule with ?₁-SVM and other common methods show very promising performance, in that the proposed method leads to much leaner models without compromising misclassification rates, particularly for high-dimensional predictors.