Support vector machine

From Wikipedia, the free encyclopedia

Support vector machines (SVMs) are a set of related supervised learning methods used for classification and regression. They belong to a family of generalized linear classifiers. They can also be considered a special case of Tikhonov regularization. A special property of SVMs is that they simultaneously minimize the empirical classification error and maximize the geometric margin. Hence it is also known as the maximum margin classifier.

Contents

[edit] Introduction

[edit] Simple Linear Classification Example

We are given a 2-dimensional grid, filled with data points. Each data point represents the location of a city near the Canada-US border. Try to draw a straight line that separates the Canadian cities from the American.

[edit] Generalization

We are not limited to geographical data, nor to two dimensions. Any piece of data that can be represented as a vector in n dimensions (e.g. an n-tuple) can be plotted and classified by a hyperplane of n-1 dimensions, just as a city on a 2-dimensional grid can be classified by a 1-dimensional line running from east to west.

[edit] Motivation

There are many linear classifiers (hyperplanes) that separate the data. However only one of these achieves maximum separation.
Enlarge
There are many linear classifiers (hyperplanes) that separate the data. However only one of these achieves maximum separation.

Often we are interested in classifying data as a part of a machine-learning process. These data points may not necessarily be points in \mathbb{R}^2 but may be multidimensional \mathbb{R}^p (statistics notation) or \mathbb{R}^n (computer science notation) points. We are interested in whether we can separate them by a n-1 dimensional hyperplane. This is a typical form of linear classifier. There are many linear classifiers that might satisfy this property. However, we are additionally interested to find out if we can achieve maximum separation (margin) between the two classes. Now, if such a hyperplane exists, the hyperplane is clearly of interest and is known as the maximum-margin hyperplane and such a linear classifier is known as a maximum margin classifier.

[edit] Formalization

Maximum-margin hyperplanes for a SVM trained with samples from two classes. Samples along the hyperplanes are called the support vectors.
Enlarge
Maximum-margin hyperplanes for a SVM trained with samples from two classes. Samples along the hyperplanes are called the support vectors.

We consider data points of the form:

\{ (\mathbf{x}_1, c_1), (\mathbf{x}_2, c_2), \ldots, (\mathbf{x}_n, c_n)\}

where the ci is either 1 or −1 -- this constant denotes the class to which the point \mathbf{x}_i belongs. Each \mathbf{x}_i is a p − (statistics notation), or n − (computer science notation) dimensional vector of scaled [0,1] or [-1,1] values. The scaling is important to guard against variables (attributes) with larger variance that might otherwise dominate the classification. We can view this as training data, which denotes the correct classification which we would like the SVM to eventually distinguish, by means of the dividing hyperplane, which takes the form

\mathbf{w}\cdot\mathbf{x} - b=0.

The vector \mathbf{w} points perpendicular to the separating hyperplane. Adding the offset parameter b allows us to increase the margin. In its absence, the hyperplane is forced to pass through the origin, restricting the solution.

As we are interested in the maximum margin, we are interested in the support vectors and the parallel hyperplanes (to the optimal hyperplane) closest to these support vectors in either class. It can be shown that these parallel hyperplanes can be described by equations

\mathbf{w}\cdot\mathbf{x} - b=1,
\mathbf{w}\cdot\mathbf{x} - b=-1.

If the training data are linearly separable, we can select these hyperlines so that there are no points between them and then try to maximize their distance. By using geometry, we find the distance between the hyperplanes is 2/|w|, so we want to minimize |w|. To exclude data points, we need to ensure that for all i either

\mathbf{w}\cdot\mathbf{x_i} - b \ge 1\qquad\mathrm{or}
\mathbf{w}\cdot\mathbf{x_i} - b \le -1\qquad\mathrm{}

This can be rewritten as:

c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1, \quad 1 \le i \le n.\qquad\qquad(1)

[edit] Primal Form

The problem now is to minimize |w| subject to the constraint (1). This is a quadratic programming (QP) optimization problem. More clearly,

minimize (1/2)||\mathbf{w}||^2, subject to c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1, \quad 1 \le i \le n..

The factor of 1/2 is used for mathematical convenience.

[edit] Dual Form

Writing the classification rule in its dual form reveals that classification is only a function of the Support vectors, i.e., the training data that lie on the margin. The dual of the SVM can be shown to be: \max \sum_{i=1}^n \alpha_i - \sum_{i,j} \alpha_i \alpha_j c_i c_j \mathbf{x}_i^T \mathbf{x}_j subject to \alpha_i \geq 0, where the α terms constitute a dual representation for the weight vector in terms of the training set: \mathbf{w} = \sum_i \alpha_i c_i \mathbf{x}_i

[edit] Soft margin

In 1995, Corinna Cortes and Vapnik suggested a modified maximum margin idea that allows for mislabeled examples. If there exists no hyperplane that can split the "yes" and "no" examples, the Soft Margin method will choose a hyperplane that splits the examples as cleanly as possible, while still maximizing the distance to the nearest cleanly split examples. This work popularized the expression Support Vector Machine or SVM. The method introduces slack variables, ξi, which measure the degree of misclassification of the datum xi

c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1 - \xi_i \quad 1 \le i \le n \quad\quad(2).

The objective function is then increased by a function which penalises non-zero ξi, and the optimisation becomes a trade off between a large margin, and a small error penalty. If the penalty function is linear, the equation (3) now transforms to

\min ||\mathbf{w}||^2 + C \sum_i \xi_i \quad \mathbf{such \; that \;}c_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1 - \xi_i \quad 1 \le i \le n

This constraint in (2) along with the objective of minimizing |w| can be solved using Lagrange multipliers. The key advantage of a linear penalty function is that the slack variables vanish from the dual problem, with the constant C appearing only as an additional constraint on the Lagrange multipliers. Non-linear penalty functions have been used, particularly to reduce the effect of outliers on the classifier, but unless care is taken, the problem becomes non-convex, and thus it is considerably more difficult to find a global solution.

[edit] Non-linear classification

The original optimal hyperplane algorithm proposed by Vladimir Vapnik in 1963 was a linear classifier. However, in 1992, Bernhard Boser, Isabelle Guyon and Vapnik suggested a way to create non-linear classifiers by applying the kernel trick (originally proposed by Aizerman) to maximum-margin hyperplanes. The resulting algorithm is formally similar, except that every dot product is replaced by a non-linear kernel function. This allows the algorithm to fit the maximum-margin hyperplane in the transformed feature space. The transformation may be non-linear and the transformed space high dimensional; thus though the classifier is a hyperplane in the high-dimensional feature space it may be non-linear in the original input space.

If the kernel used is a Gaussian radial basis function, the corresponding feature space is a Hilbert space of infinite dimension. Maximum margin classifiers are well regularized, so the infinite dimension does not spoil the results. Some common kernels include,

  • Polynomial (homogeneous): k(\mathbf{x},\mathbf{x}')=(\mathbf{x} \cdot \mathbf{x'})^d
  • Polynomial (inhomogeneous): k(\mathbf{x},\mathbf{x}')=(\mathbf{x} \cdot \mathbf{x'} + 1)^d
  • Radial Basis Function: k(\mathbf{x},\mathbf{x}')=\exp(-\gamma \|\mathbf{x} - \mathbf{x'}\|^2), for γ > 0
  • Gaussian Radial basis function: k(\mathbf{x},\mathbf{x}')=\exp(- \frac{\|\mathbf{x} - \mathbf{x'}\|^2}{2 \sigma^2})
  • Sigmoid: k(\mathbf{x},\mathbf{x}')=\tanh(\kappa \mathbf{x} \cdot \mathbf{x'}+c), for some (not every) κ > 0 and c < 0

[edit] Regression

A version of a SVM for regression was proposed in 1996 by Vapnik, Harris Drucker, Chris Burges, Linda Kaufman and Alex Smola. This method is called support vector regression (SVR). The model produced by support vector classification (as described above) only depends on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin. Analogously, the model produced by SVR only depends on a subset of the training data, because the cost function for building the model ignores any training data that are close (within a threshold ε) to the model prediction.

[edit] Implementation

The parameters of the maximum-margin hyperplane are derived by solving the optimization. There exist several specialized algorithms for quickly solving the QP problem that arises from SVMs, mostly reliant on heuristics for breaking the problem down into smaller, more-manageble chunks. A common method for solving the QP problem is Platt's SMO algorithm, which breaks the problem down into 2-dimensional sub-problems that may be solved analytically, eliminating the need for a numerical optimization algorithm such as conjugate gradient methods.

[edit] See also

[edit] References

  • B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, 5th Annual ACM Workshop on COLT, pages 144-152, Pittsburgh, PA, 1992. ACM Press.
  • Corinna Cortes and V. Vapnik, "Support-Vector Networks, Machine Learning, 20, 1995. [1]
  • Christopher J. C. Burges. "A Tutorial on Support Vector Machines for Pattern Recognition". Data Mining and Knowledge Discovery 2:121 - 167, 1998 (Also available at CiteSeer: [2])
  • Nello Cristianini and John Shawe-Taylor. An Introduction to Support Vector Machines and other kernel-based learning methods. Cambridge University Press, 2000. ISBN 0-521-78019-5 ([3] SVM Book)
  • Harris Drucker, Chris J.C. Burges, Linda Kaufman, Alex Smola and Vladimir Vapnik (1997). "Support Vector Regression Machines". Advances in Neural Information Processing Systems 9, NIPS 1996, 155-161, MIT Press.
  • Huang T.-M., Kecman V., Kopriva I. (2006), Kernel Based Algorithms for Mining Huge Data Sets, Supervised, Semi-supervised, and Unsupervised Learning, Springer-Verlag, Berlin, Heidelberg, 260 pp. 96 illus., Hardcover, ISBN 3-540-31681-7[4]
  • Vojislav Kecman: "Learning and Soft Computing - Support Vector Machines, Neural Networks, Fuzzy Logic Systems", The MIT Press, Cambridge, MA, 2001.[5]
  • Tapio Pahikkala, Sampo Pyysalo, Jorma Boberg, Aleksandr Mylläri and Tapio Salakoski. Improving the Performance of Bayesian and Support Vector Classifiers in Word Sense Disambiguation using Positional Information. In Proceedings of the International and Interdisciplinary Conference on Adaptive Knowledge Representation and Reasoning (AKRR'05), Jun 2005.
  • Bernhard Schölkopf and A. J. Smola: Learning with Kernels. MIT Press, Cambridge, MA, 2002. (Partly available on line: [6].) ISBN 0-262-19475-9
  • Bernhard Schölkopf, Christopher J.C. Burges, and Alexander J. Smola (editors). "Advances in Kernel Methods: Support Vector Learning". MIT Press, Cambridge, MA, 1999. ISBN 0-262-19416-3. [7]
  • John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. ISBN 0-521-81397-2 ([8] Kernel Methods Book)
  • P.J. Tan and D.L. Dowe (2004), MML Inference of Oblique Decision Trees, Lecture Notes in Artificial Intelligence (LNAI) 3339, Springer-Verlag, pp1082-1088. Links require password. (This paper uses minimum message length (MML) and actually incorporates probabilistic support vector machines in the leaves of decision trees.)
  • Vladimir Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag, 1999. ISBN 0-387-98780-0

[edit] External links

[edit] General

[edit] Software

  • Lush -- a Lisp-like interpreted/compiled language with C/C++/Fortran interfaces that has packages to interface to a number of different SVM implementations. Interfaces to LASVM, LIBSVM, mySVM, SVQP, SVQP2 (SVQP3 in future) are available. Leverage these against Lush's other interfaces to machine learning, hidden markov models, numerical libraries (LAPACK, BLAS, GSL), and builtin vector/matrix/tensor engine.
  • SVMlight -- a popular implementation of the SVM algorithm by Thorsten Joachims; it can be used to solve classification, regression and ranking problems.
  • LIBSVM -- A Library for Support Vector Machines, Chih-Chung Chang and Chih-Jen Lin
  • YALE -- a powerful machine learning toolbox containing wrappers for SVMLight, LibSVM, and MySVM in addition to many evaluation and preprocessing methods.
  • LS-SVMLab - Matlab/C SVM toolbox - well-documented, many features
  • Gist -- implementation of the SVM algorithm with feature selection.
  • Weka -- a machine learning toolkit that includes an implementation of an SVM classifier; Weka can be used both interactively though a graphical interface or as a software library. (One of them is called "SMO". In the GUI Weka explorer, it is under the "classify" tab if you "Choose" an algorithm.)
  • OSU SVM - Matlab implementation based on LIBSVM
  • Torch - C++ machine learning library with SVM
  • Shogun - Large Scale Machine Learning Toolbox with interfaces to Octave, Matlab, Python, R
  • Spider - Machine learning library for Matlab
  • e1071 - Machine learning library for R
  • SimpleSVM - SimpleSVM toolbox for Matlab
  • SVM and Kernel Methods Matlab Toolbox
  • PCP -- C program for supervised pattern classification. Includes LIBSVM wrapper.
  • TinySVM -- a small SVM implementation, written in C++

[edit] Interactive SVM applications

Retrieved from "http://en.wikipedia.org../../../s/u/p/Support_vector_machine.html"