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44
A tutorial on support vector regression
, 2004
"... In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing ..."
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Cited by 473 (2 self)
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In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.
Sparse Greedy Matrix Approximation for Machine Learning
, 2000
"... In kernel based methods such as Regularization Networks large datasets pose signi cant problems since the number of basis functions required for an optimal solution equals the number of samples. We present a sparse greedy approximation technique to construct a compressed representation of the ..."
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Cited by 179 (11 self)
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In kernel based methods such as Regularization Networks large datasets pose signi cant problems since the number of basis functions required for an optimal solution equals the number of samples. We present a sparse greedy approximation technique to construct a compressed representation of the design matrix. Experimental results are given and connections to KernelPCA, Sparse Kernel Feature Analysis, and Matching Pursuit are pointed out. 1. Introduction Many recent advances in machine learning such as Support Vector Machines [Vapnik, 1995], Regularization Networks [Girosi et al., 1995], or Gaussian Processes [Williams, 1998] are based on kernel methods. Given an msample f(x 1 ; y 1 ); : : : ; (x m ; y m )g of patterns x i 2 X and target values y i 2 Y these algorithms minimize the regularized risk functional min f2H R reg [f ] = 1 m m X i=1 c(x i ; y i ; f(x i )) + 2 kfk 2 H : (1) Here H denotes a reproducing kernel Hilbert space (RKHS) [Aronszajn, 1950],...
Linear programming boosting via column generation
 Machine Learning
, 2002
"... 1 Introduction Recent papers [20] have shown that boosting, arcing, and related ensemble methods (hereafter summarized asboosting) can be viewed as margin maximization in function space. By changing the cost function, different ..."
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Cited by 101 (3 self)
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1 Introduction Recent papers [20] have shown that boosting, arcing, and related ensemble methods (hereafter summarized asboosting) can be viewed as margin maximization in function space. By changing the cost function, different
Pattern Recognition Via Linear Programming: Theory And Application To Medical Diagnosis
, 1990
"... . A decision problem associated with a fundamental nonconvex model for linearly inseparable pattern sets is shown to be NPcomplete. Another nonconvex model that employs an 1\Gamma norm instead of the 2norm, can be solved in polynomial time by solving 2n linear programs, where n is the (usually sm ..."
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Cited by 69 (13 self)
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. A decision problem associated with a fundamental nonconvex model for linearly inseparable pattern sets is shown to be NPcomplete. Another nonconvex model that employs an 1\Gamma norm instead of the 2norm, can be solved in polynomial time by solving 2n linear programs, where n is the (usually small) dimensionality of the pattern space. An effective LPbased finite algorithm is proposed for solving the latter model. The algorithm is employed to obtain a nonconvex piecewiselinear function for separating points representing measurements made on fine needle aspirates taken from benign and malignant human breasts. A computer program trained on 369 samples has correctly diagnosed each of 45 new samples encountered and is currently in use at the University of Wisconsin Hospitals. 1. Introduction. The fundamental problem we wish to address is that of distinguishing between elements of two distinct pattern sets. Mathematically we can formulate the problem as follows. Given two disjoint fin...
Duality and Geometry in SVM Classifiers
 In Proc. 17th International Conf. on Machine Learning
, 2000
"... We develop an intuitive geometric interpretation of the standard support vector machine (SVM) for classification of both linearly separable and inseparable data and provide a rigorous derivation of the concepts behind the geometry. For the separable case finding the maximum margin between the ..."
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Cited by 59 (4 self)
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We develop an intuitive geometric interpretation of the standard support vector machine (SVM) for classification of both linearly separable and inseparable data and provide a rigorous derivation of the concepts behind the geometry. For the separable case finding the maximum margin between the two sets is equivalent to finding the closest points in the smallest convex sets that contain each class (the convex hulls). We now extend this argument to the inseparable case by using a reduced convex hull reduced away from outliers. We prove that solving the reduced convex hull formulation is exactly equivalent to solving the standard inseparable SVM for appropriate choices of parameters. Some additional advantages of the new formulation are that the e#ect of the choice of parameters becomes geometrically clear and that the formulation may be solved by fast nearest point algorithms. By changing norms these arguments hold for both the standard 2norm and 1norm SVM. 1. Int...
A scalable modular convex solver for regularized risk minimization
 In KDD. ACM
, 2007
"... A wide variety of machine learning problems can be described as minimizing a regularized risk functional, with different algorithms using different notions of risk and different regularizers. Examples include linear Support Vector Machines (SVMs), Logistic Regression, Conditional Random Fields (CRFs ..."
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Cited by 59 (14 self)
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A wide variety of machine learning problems can be described as minimizing a regularized risk functional, with different algorithms using different notions of risk and different regularizers. Examples include linear Support Vector Machines (SVMs), Logistic Regression, Conditional Random Fields (CRFs), and Lasso amongst others. This paper describes the theory and implementation of a highly scalable and modular convex solver which solves all these estimation problems. It can be parallelized on a cluster of workstations, allows for datalocality, and can deal with regularizers such as ℓ1 and ℓ2 penalties. At present, our solver implements 20 different estimation problems, can be easily extended, scales to millions of observations, and is up to 10 times faster than specialized solvers for many applications. The open source code is freely available as part of the ELEFANT toolbox.
Multicategory Classification by Support Vector Machines
 Computational Optimizations and Applications
, 1999
"... We examine the problem of how to discriminate between objects of three or more classes. Specifically, we investigate how twoclass discrimination methods can be extended to the multiclass case. We show how the linear programming (LP) approaches based on the work of Mangasarian and quadratic programm ..."
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Cited by 56 (0 self)
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We examine the problem of how to discriminate between objects of three or more classes. Specifically, we investigate how twoclass discrimination methods can be extended to the multiclass case. We show how the linear programming (LP) approaches based on the work of Mangasarian and quadratic programming (QP) approaches based on Vapnik's Support Vector Machines (SVM) can be combined to yield two new approaches to the multiclass problem. In LP multiclass discrimination, a single linear program is used to construct a piecewise linear classification function. In our proposed multiclass SVM method, a single quadratic program is used to construct a piecewise nonlinear classification function. Each piece of this function can take the form of a polynomial, radial basis function, or even a neural network. For the k > 2 class problems, the SVM method as originally proposed required the construction of a twoclass SVM to separate each class from the remaining classes. Similarily, k twoclass linear programs can be used for the multiclass problem. We performed an empirical study of the original LP method, the proposed k LP method, the proposed single QP method and the original k QP methods. We discuss the advantages and disadvantages of each approach. 1 1
Massive Data Discrimination via Linear Support Vector Machines
 Optimization Methods and Software
, 1998
"... A linear support vector machine formulation is used to generate a fast, finitelyterminating linearprogramming algorithm for discriminating between two massive sets in ndimensional space, where the number of points can be orders of magnitude larger than n. The algorithm creates a succession of su ..."
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Cited by 48 (16 self)
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A linear support vector machine formulation is used to generate a fast, finitelyterminating linearprogramming algorithm for discriminating between two massive sets in ndimensional space, where the number of points can be orders of magnitude larger than n. The algorithm creates a succession of sufficiently small linear programs that separate chunks of the data at a time. The key idea is that a small number of support vectors, corresponding to linear programming constraints with positive dual variables, are carried over between the successive small linear programs, each of which containing a chunk of the data. We prove that this procedure is monotonic and terminates in a finite number of steps at an exact solution that leads to a globally optimal separating plane for the entire dataset. Numerical results on fully dense publicly available datasets, numbering 20,000 to 1 million points in 32dimensional space, confirm the theoretical results and demonstrate the ability to handle very l...
Mathematical Programming for Data Mining: Formulations and Challenges
 INFORMS Journal on Computing
, 1998
"... This paper is intended to serve as an overview of a rapidly emerging research and applications area. In addition to providing a general overview, motivating the importance of data mining problems within the area of knowledge discovery in databases, our aim is to list some of the pressing research ch ..."
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Cited by 47 (0 self)
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This paper is intended to serve as an overview of a rapidly emerging research and applications area. In addition to providing a general overview, motivating the importance of data mining problems within the area of knowledge discovery in databases, our aim is to list some of the pressing research challenges, and outline opportunities for contributions by the optimization research communities. Towards these goals, we include formulations of the basic categories of data mining methods as optimization problems. We also provide examples of successful mathematical programming approaches to some data mining problems. keywords: data analysis, data mining, mathematical programming methods, challenges for massive data sets, classification, clustering, prediction, optimization. To appear: INFORMS: Journal of Compting, special issue on Data Mining, A. Basu and B. Golden (guest editors). Also appears as Mathematical Programming Technical Report 9801, Computer Sciences Department, University of Wi...
Mathematical Programming in Neural Networks
 ORSA Journal on Computing
, 1993
"... This paper highlights the role of mathematical programming, particularly linear programming, in training neural networks. A neural network description is given in terms of separating planes in the input space that suggests the use of linear programming for determining these planes. A more standard d ..."
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Cited by 40 (13 self)
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This paper highlights the role of mathematical programming, particularly linear programming, in training neural networks. A neural network description is given in terms of separating planes in the input space that suggests the use of linear programming for determining these planes. A more standard description in terms of a mean square error in the output space is also given, which leads to the use of unconstrained minimization techniques for training a neural network. The linear programming approach is demonstrated by a brief description of a system for breast cancer diagnosis that has been in use for the last four years at a major medical facility. 1 What is a Neural Network? A neural network is a representation of a map between an input space and an output space. A principal aim of such a map is to discriminate between the elements of a finite number of disjoint sets in the input space. Typically one wishes to discriminate between the elements of two disjoint point sets in the ndim...