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75
A tutorial on support vector machines for pattern recognition
 Data Mining and Knowledge Discovery
, 1998
"... The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and nonseparable data, working through a nontrivial example in detail. We describe a mechanical analogy, and discuss when SV ..."
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Cited by 2497 (11 self)
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The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and nonseparable data, working through a nontrivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.
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 540 (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.
Kernel principal component analysis
 ADVANCES IN KERNEL METHODS  SUPPORT VECTOR LEARNING
, 1999
"... A new method for performing a nonlinear form of Principal Component Analysis is proposed. By the use of integral operator kernel functions, one can efficiently compute principal components in highdimensional feature spaces, related to input space by some nonlinear map; for instance the space of all ..."
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Cited by 189 (7 self)
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A new method for performing a nonlinear form of Principal Component Analysis is proposed. By the use of integral operator kernel functions, one can efficiently compute principal components in highdimensional feature spaces, related to input space by some nonlinear map; for instance the space of all possible dpixel products in images. We give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition.
Improving the Accuracy and Speed of Support Vector Machines
 Advances in Neural Information Processing Systems 9
, 1997
"... Support Vector Learning Machines (SVM) are finding application in pattern recognition, regression estimation, and operator inversion for illposed problems. Against this very general backdrop, any methods for improving the generalization performance, or for improving the speed in test phase, of SVMs ..."
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Cited by 153 (20 self)
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Support Vector Learning Machines (SVM) are finding application in pattern recognition, regression estimation, and operator inversion for illposed problems. Against this very general backdrop, any methods for improving the generalization performance, or for improving the speed in test phase, of SVMs are of increasing interest. In this paper we combine two such techniques on a pattern recognition problem. The method for improving generalization performance (the "virtual support vector" method) does so by incorporating known invariances of the problem. This method achieves a drop in the error rate on 10,000 NIST test digit images of 1.4% to 1.0%. The method for improving the speed (the "reduced set" method) does so by approximating the support vector decision surface. We apply this method to achieve a factor of fifty speedup in test phase over the virtual support vector machine. The combined approach yields a machine which is both 22 times faster than the original machine, and which has ...
Training Invariant Support Vector Machines
, 2002
"... Practical experience has shown that in order to obtain the best possible performance, prior knowledge about invariances of a classification problem at hand ought to be incorporated into the training procedure. We describe and review all known methods for doing so in support vector machines, provide ..."
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Cited by 149 (16 self)
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Practical experience has shown that in order to obtain the best possible performance, prior knowledge about invariances of a classification problem at hand ought to be incorporated into the training procedure. We describe and review all known methods for doing so in support vector machines, provide experimental results, and discuss their respective merits. One of the significant new results reported in this work is our recent achievement of the lowest reported test error on the wellknown MNIST digit recognition benchmark task, with SVM training times that are also significantly faster than previous SVM methods.
Comparing Support Vector Machines with Gaussian Kernels to Radial Basis Function Classifiers
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 1997
"... The Support Vector (SV) machine is a novel type of learning machine, based on statistical learning theory, which contains polynomial classifiers, neural networks, and radial basis function (RBF) networks as special cases. In the RBF case, the SV algorithm automatically determines centers, weights an ..."
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Cited by 136 (12 self)
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The Support Vector (SV) machine is a novel type of learning machine, based on statistical learning theory, which contains polynomial classifiers, neural networks, and radial basis function (RBF) networks as special cases. In the RBF case, the SV algorithm automatically determines centers, weights and threshold such as to minimize an upper bound on the expected test error. The present study is devoted to an experimental comparison of these machines with a classical approach, where the centers are determined by kmeans clustering and the weights are found using error backpropagation. We consider three machines, namely a classical RBF machine, an SV machine with Gaussian kernel, and a hybrid system with the centers determined by the SV method and the weights trained by error backpropagation. Our results show that on the US postal service database of handwritten digits, the SV machine achieves the highest test accuracy, followed by the hybrid approach. The SV approach is thus not only the...
Prior Knowledge in Support Vector Kernels
, 1998
"... We explore methods for incorporating prior knowledge about a problem at hand in Support Vector learning machines. We show that both invariances under group transformations and prior knowledge about locality in images can be incorporated by constructing appropriate kernel functions. ..."
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Cited by 105 (13 self)
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We explore methods for incorporating prior knowledge about a problem at hand in Support Vector learning machines. We show that both invariances under group transformations and prior knowledge about locality in images can be incorporated by constructing appropriate kernel functions.
On a Kernelbased Method for Pattern Recognition, Regression, Approximation, and Operator Inversion
, 1997
"... We present a Kernelbased framework for Pattern Recognition, Regression Estimation, Function Approximation and multiple Operator Inversion. Previous approaches such as ridgeregression, Support Vector methods and regression by Smoothing Kernels are included as special cases. We will show connection ..."
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Cited by 79 (23 self)
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We present a Kernelbased framework for Pattern Recognition, Regression Estimation, Function Approximation and multiple Operator Inversion. Previous approaches such as ridgeregression, Support Vector methods and regression by Smoothing Kernels are included as special cases. We will show connections between the costfunction and some properties up to now believed to apply to Support Vector Machines only. The optimal solution of all the problems described above can be found by solving a simple quadratic programming problem. The paper closes with a proof of the equivalence between Support Vector kernels and Greene's functions of regularization operators.
Moderating the Outputs of Support Vector Machine Classifiers
 IEEE Transactions on Neural Networks
, 1999
"...  In this paper, we extend the use of moderated outputs to the support vector machine (SVM) by making use of a relationship between SVM and the evidence framework. The moderated output is more in line with the Bayesian idea that the posterior weight distribution should be taken into account upon pre ..."
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Cited by 46 (3 self)
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 In this paper, we extend the use of moderated outputs to the support vector machine (SVM) by making use of a relationship between SVM and the evidence framework. The moderated output is more in line with the Bayesian idea that the posterior weight distribution should be taken into account upon prediction, and it also alleviates the usual tendency of assigning overly high condence to the estimated class memberships of the test patterns. Moreover, the moderated output derived here can be taken as an approximation to the posterior class probability. Hence, meaningful rejection thresholds can be assigned and outputs from several networks can be directly compared. Experimental results on both articial and realworld data are also discussed. KeywordsSupport vector machine, Evidence framework, Moderated output, Bayesian I. Introduction I N recent years, there has been a lot of interest in studying the support vector machine (SVM) [1], [2], [3], [4], [5], [6], [7]. SVM is based on the i...