The analysis of decomposition methods for support vector machines
TL;DR: This paper connects this method to projected gradient methods and provides theoretical proofs for a version of decomposition methods and shows that this convergence proof is valid for general decomposition Methods if their working set selection meets a simple requirement.
Abstract: The support vector machine (SVM) is a promising technique for pattern recognition. It requires the solution of a large dense quadratic programming problem. Traditional optimization methods cannot be directly applied due to memory restrictions. Up to now, very few methods can handle the memory problem and an important one is the "decomposition method." However, there is no convergence proof so far. We connect this method to projected gradient methods and provide theoretical proofs for a version of decomposition methods. An extension to bound-constrained formulation of SVM is also provided. We then show that this convergence proof is valid for general decomposition methods if their working set selection meets a simple requirement.
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TL;DR: Issues such as solving SVM optimization problems theoretical convergence multiclass classification probability estimates and parameter selection are discussed in detail.
Abstract: LIBSVM is a library for Support Vector Machines (SVMs). We have been actively developing this package since the year 2000. The goal is to help users to easily apply SVM to their applications. LIBSVM has gained wide popularity in machine learning and many other areas. In this article, we present all implementation details of LIBSVM. Issues such as solving SVM optimization problems theoretical convergence multiclass classification probability estimates and parameter selection are discussed in detail.
40,826 citations
Cites methods from "The analysis of decomposition metho..."
...The convergence of decomposition methods was first studied in (Chang et al., 2000) but algorithms discussed there do not coincide with existing implementations....
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...However, its result applies only to decomposition methods discussed in (Chang et al., 2000) but not LIBSVM or other existing software....
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TL;DR: This tutorial gives an overview of the basic ideas underlying Support Vector (SV) machines for function estimation, and includes a summary of currently used algorithms for training SV machines, covering both the quadratic programming part and advanced methods for dealing with large datasets.
Abstract: 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.
10,696 citations
Cites background from "The analysis of decomposition metho..."
...Still in practice one has to take special precautions to avoid stalling of convergence (recent results of Chang et al. [1999] indicate that under certain conditions a proof of convergence is possible)....
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TL;DR: The ability of SVM to outperform several well-known methods developed for the widely studied problem of MC detection suggests that SVM is a promising technique for object detection in a medical imaging application.
Abstract: We investigate an approach based on support vector machines (SVMs) for detection of microcalcification (MC) clusters in digital mammograms, and propose a successive enhancement learning scheme for improved performance. SVM is a machine-learning method, based on the principle of structural risk minimization, which performs well when applied to data outside the training set. We formulate MC detection as a supervised-learning problem and apply SVM to develop the detection algorithm. We use the SVM to detect at each location in the image whether an MC is present or not. We tested the proposed method using a database of 76 clinical mammograms containing 1120 MCs. We use free-response receiver operating characteristic curves to evaluate detection performance, and compare the proposed algorithm with several existing methods. In our experiments, the proposed SVM framework outperformed all the other methods tested. In particular, a sensitivity as high as 94% was achieved by the SVM method at an error rate of one false-positive cluster per image. The ability of SVM to outperform several well-known methods developed for the widely studied problem of MC detection suggests that SVM is a promising technique for object detection in a medical imaging application.
574 citations
Cites methods from "The analysis of decomposition metho..."
...In this paper, we adopted a technique called successive minimal optimization (SMO) [30]–[32]....
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TL;DR: A decomposition method for -SVM is proposed that is competitive with existing methods for C-SVM and shows that in general they are two different problems with the same optimal solution set.
Abstract: The ν-support vector machine (ν-SVM) for classification proposed by Scholkopf, Smola, Williamson, and Bartlett (2000) has the advantage of using a parameter ν on controlling the number of support vectors. In this article, we investigate the relation between ν-SVM and C-SVM in detail. We show that in general they are two different problems with the same optimal solution set. Hence, we may expect that many numerical aspects of solving them are similar. However, compared to regular C-SVM, the formulation of ν-SVM is more complicated, so up to now there have been no effective methods for solving large-scale ν-SVM. We propose a decomposition method for ν-SVM that is competitive with existing methods for C-SVM. We also discuss the behavior of ν-SVM by some numerical experiments.
461 citations
Cites background from "The analysis of decomposition metho..."
...This was first pointed out by Chang et al. (2000)....
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...The strict decrease of the objective function holds, and the theoretical convergence was studied in Chang, Hsu, and Lin (2000), Keerthi and Gilbert (2000), and Lin (2000)....
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01 Jan 2007
TL;DR: This chapter contains sections titled: Introduction, Support Vector Machines, Duality, Sparsity, Early SVM Algorithms, The Decomposition Method, A Case Study: LIBSVM, Conclusion and Outlook.
Abstract: This chapter contains sections titled: Introduction, Support Vector Machines, Duality, Sparsity, Early SVM Algorithms, The Decomposition Method, A Case Study: LIBSVM, Conclusion and Outlook, Appendix
324 citations
Additional excerpts
...With suitable working set selection schemes, asymptotic convergence results state that any limit point of the infinite sequence generated by the algorithm is an optimal solution (e.g. Chang et al., 2000; Lin, 2001; Hush and Scovel, 2003; List and Simon, 2004; Palagi and Sciandrone, 2005)....
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References
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TL;DR: High generalization ability of support-vector networks utilizing polynomial input transformations is demonstrated and the performance of the support- vector network is compared to various classical learning algorithms that all took part in a benchmark study of Optical Character Recognition.
Abstract: The support-vector network is a new learning machine for two-group classification problems. The machine conceptually implements the following idea: input vectors are non-linearly mapped to a very high-dimension feature space. In this feature space a linear decision surface is constructed. Special properties of the decision surface ensures high generalization ability of the learning machine. The idea behind the support-vector network was previously implemented for the restricted case where the training data can be separated without errors. We here extend this result to non-separable training data.
High generalization ability of support-vector networks utilizing polynomial input transformations is demonstrated. We also compare the performance of the support-vector network to various classical learning algorithms that all took part in a benchmark study of Optical Character Recognition.
37,861 citations
01 Jan 1998
TL;DR: Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.
Abstract: A comprehensive look at learning and generalization theory. The statistical theory of learning and generalization concerns the problem of choosing desired functions on the basis of empirical data. Highly applicable to a variety of computer science and robotics fields, this book offers lucid coverage of the theory as a whole. Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.
26,531 citations
"The analysis of decomposition metho..." refers methods in this paper
...Surveys of SVM are, for example, Burges [1], Cortes and Vapnik [2], Scholkopf et al. [3], and Vapnik [ 4 ]....
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TL;DR: There are several arguments which support the observed high accuracy of SVMs, which are reviewed and numerous examples and proofs of most of the key theorems are given.
Abstract: 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 non-separable data, working through a non-trivial 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.
15,696 citations
TL;DR: This tutorial gives an overview of the basic ideas underlying Support Vector (SV) machines for function estimation, and includes a summary of currently used algorithms for training SV machines, covering both the quadratic programming part and advanced methods for dealing with large datasets.
Abstract: 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.
10,696 citations