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Making LargeScale Support Vector Machine Learning Practical
, 1998
"... Training a support vector machine (SVM) leads to a quadratic optimization problem with bound constraints and one linear equality constraint. Despite the fact that this type of problem is well understood, there are many issues to be considered in designing an SVM learner. In particular, for large lea ..."
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Cited by 628 (1 self)
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Training a support vector machine (SVM) leads to a quadratic optimization problem with bound constraints and one linear equality constraint. Despite the fact that this type of problem is well understood, there are many issues to be considered in designing an SVM learner. In particular, for large learning tasks with many training examples, offtheshelf optimization techniques for general quadratic programs quickly become intractable in their memory and time requirements. SVM light1 is an implementation of an SVM learner which addresses the problem of large tasks. This chapter presents algorithmic and computational results developed for SVM light V2.0, which make largescale SVM training more practical. The results give guidelines for the application of SVMs to large domains.
An introduction to kernelbased learning algorithms
 IEEE TRANSACTIONS ON NEURAL NETWORKS
, 2001
"... This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and ..."
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Cited by 598 (55 self)
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This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and
Support Vector Machine Classification and Validation of Cancer Tissue Samples Using Microarray Expression Data
, 2000
"... Motivation: DNA microarray experiments generating thousands of gene expression measurements, are being used to gather information from tissue and cell samples regarding gene expression differences that will be useful in diagnosing disease. We have developed a new method to analyse this kind of data ..."
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Cited by 569 (1 self)
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Motivation: DNA microarray experiments generating thousands of gene expression measurements, are being used to gather information from tissue and cell samples regarding gene expression differences that will be useful in diagnosing disease. We have developed a new method to analyse this kind of data using support vector machines (SVMs). This analysis consists of both classification of the tissue samples, and an exploration of the data for mislabeled or questionable tissue results. Results: We demonstrate the method in detail on samples consisting of ovarian cancer tissues, normal ovarian tissues, and other normal tissues. The dataset consists of expression experiment results for 97 802 cDNAs for each tissue. As a result of computational analysis, a tissue sample is discovered and confirmed to be wrongly labeled. Upon correction of this mistake and the removal of an outlier, perfect classification of tissues is achieved, but not with high confidence. We identify and analyse a subset of genes from the ovarian dataset whose expression is highly differentiated between the types of tissues. To show robustness of the SVM method, two previously published datasets from other types of tissues or cells are analysed. The results are comparable to those previously obtained. We show that other machine learning methods also perform comparably to the SVM on many of those datasets. Availability: The SVM software is available at http:// www. cs.columbia.edu/#bgrundy/svm. Contact: booch@cse.ucsc.edu
On the algorithmic implementation of multiclass kernelbased vector machines
 Journal of Machine Learning Research
"... In this paper we describe the algorithmic implementation of multiclass kernelbased vector machines. Our starting point is a generalized notion of the margin to multiclass problems. Using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic ob ..."
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Cited by 559 (13 self)
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In this paper we describe the algorithmic implementation of multiclass kernelbased vector machines. Our starting point is a generalized notion of the margin to multiclass problems. Using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic objective function. Unlike most of previous approaches which typically decompose a multiclass problem into multiple independent binary classification tasks, our notion of margin yields a direct method for training multiclass predictors. By using the dual of the optimization problem we are able to incorporate kernels with a compact set of constraints and decompose the dual problem into multiple optimization problems of reduced size. We describe an efficient fixedpoint algorithm for solving the reduced optimization problems and prove its convergence. We then discuss technical details that yield significant running time improvements for large datasets. Finally, we describe various experiments with our approach comparing it to previously studied kernelbased methods. Our experiments indicate that for multiclass problems we attain stateoftheart accuracy.
Large Margin Classification Using the Perceptron Algorithm
 Machine Learning
, 1998
"... We introduce and analyze a new algorithm for linear classification which combines Rosenblatt 's perceptron algorithm with Helmbold and Warmuth's leaveoneout method. Like Vapnik 's maximalmargin classifier, our algorithm takes advantage of data that are linearly separable with large ..."
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Cited by 521 (2 self)
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We introduce and analyze a new algorithm for linear classification which combines Rosenblatt 's perceptron algorithm with Helmbold and Warmuth's leaveoneout method. Like Vapnik 's maximalmargin classifier, our algorithm takes advantage of data that are linearly separable with large margins. Compared to Vapnik's algorithm, however, ours is much simpler to implement, and much more efficient in terms of computation time. We also show that our algorithm can be efficiently used in very high dimensional spaces using kernel functions. We performed some experiments using our algorithm, and some variants of it, for classifying images of handwritten digits. The performance of our algorithm is close to, but not as good as, the performance of maximalmargin classifiers on the same problem, while saving significantly on computation time and programming effort. 1 Introduction One of the most influential developments in the theory of machine learning in the last few years is Vapnik's work on supp...
Text Classification using String Kernels
"... We propose a novel approach for categorizing text documents based on the use of a special kernel. The kernel is an inner product in the feature space generated by all subsequences of length k. A subsequence is any ordered sequence of k characters occurring in the text though not necessarily contiguo ..."
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Cited by 495 (7 self)
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We propose a novel approach for categorizing text documents based on the use of a special kernel. The kernel is an inner product in the feature space generated by all subsequences of length k. A subsequence is any ordered sequence of k characters occurring in the text though not necessarily contiguously. The subsequences are weighted by anexponentially decaying factor of their full length in the text, hence emphasising those occurrences that are close to contiguous. A direct computation of this feature vector would involve a prohibitive amount of computation even for modest values of k, since the dimension of the feature space grows exponentially with k. The paper describes how despite this fact the inner product can be e ciently evaluated by a dynamic programming technique. Experimental comparisons of the performance of the kernel compared with a standard word feature space kernel Joachims (1998) show positive results on modestly sized datasets. The case of contiguous subsequences is also considered for comparison with the subsequences kernel with di erent decay factors. For larger documents and datasets the paper introduces an approximation technique that is shown to deliver good approximations e ciently for large datasets.
Sequential minimal optimization: A fast algorithm for training support vector machines
 Advances in Kernel MethodsSupport Vector Learning
, 1999
"... This paper proposes a new algorithm for training support vector machines: Sequential Minimal Optimization, or SMO. Training a support vector machine requires the solution of a very large quadratic programming (QP) optimization problem. SMO breaks this large QP problem into a series of smallest possi ..."
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Cited by 461 (3 self)
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This paper proposes a new algorithm for training support vector machines: Sequential Minimal Optimization, or SMO. Training a support vector machine requires the solution of a very large quadratic programming (QP) optimization problem. SMO breaks this large QP problem into a series of smallest possible QP problems. These small QP problems are solved analytically, which avoids using a timeconsuming numerical QP optimization as an inner loop. The amount of memory required for SMO is linear in the training set size, which allows SMO to handle very large training sets. Because matrix computation is avoided, SMO scales somewhere between linear and quadratic in the training set size for various test problems, while the standard chunking SVM algorithm scales somewhere between linear and cubic in the training set size. SMO’s computation time is dominated by SVM evaluation, hence SMO is fastest for linear SVMs and sparse data sets. On realworld sparse data sets, SMO can be more than 1000 times faster than the chunking algorithm. 1.
Classification by pairwise coupling
, 1998
"... We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then coupling the estimates together. The coupling model is similar to the BradleyTerry method for paired comparisons. We study the nature of the class probability estim ..."
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Cited by 378 (0 self)
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We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then coupling the estimates together. The coupling model is similar to the BradleyTerry method for paired comparisons. We study the nature of the class probability estimates that arise, and examine the performance of the procedure in real and simulated datasets. Classifiers used include linear discriminants, nearest neighbors, and the support vector machine.
An Improved Training Algorithm for Support Vector Machines
, 1997
"... We investigate the problem of training a Support Vector Machine (SVM) [1, 2, 7] on a very large date base (e.g. 50,000 data points) in the case in which the number of support vectors is also very large (e.g. 40,000). Training a SVM is equivalent to solving a linearly constrained quadratic programmin ..."
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Cited by 339 (1 self)
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We investigate the problem of training a Support Vector Machine (SVM) [1, 2, 7] on a very large date base (e.g. 50,000 data points) in the case in which the number of support vectors is also very large (e.g. 40,000). Training a SVM is equivalent to solving a linearly constrained quadratic programming (QP) problem in a number of variables equal to the number of data points. This optimization problem is known to be challenging when the number of data points exceeds few thousands. In previous work, done by us as well as by other researchers, the strategy used to solve the large scale QP problem takes advantage of the fact that the expected number of support vectors is small (! 3; 000). Therefore, the existing algorithms cannot deal with more than a few thousand support vectors. In this paper we present a decomposition algorithm that is guaranteed to solve the QP problem and that does not make assumptions on the expected number of support vectors. In order to present the feasibility of our...