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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 this family of classifiers is to simultaneously minimize the empirical classification error and maximize the geometric margin. Hence it is also known as maximum margin classifier.
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Motivation
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Often we are interested in classifying data as a part of a machine-learning process. These data points may not necessarily be points in but may be multidimensional (statistics notation) or (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.
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Formalization
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We consider data points of the form:
where the ci is either 1 or −1 -- this constant denotes the class to which the point belongs. Each is a (statistics notation), or (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
The vector 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
If the training data is linearly separable, we can select these hyperlines so that there is 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 either
This can be rewritten as:
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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 , subject to .
The factor of 1/2 is used for mathematical convenience.
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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.
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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, , which measure the degree of misclassification of the datum
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The objective function is then increased by a function which penalises non-zero , 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
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.
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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):
Polynomial (inhomogeneous):
Radial Basis Function: , for
Gaussian Radial basis function:
Sigmoid: , for some (not every) and
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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 is close (within a threshold ) to the model prediction.
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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, 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.
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See also
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General
www.pascal-network.org (EU Funded Network on Pattern Analysis, Statistical Modelling and Computational Learning)
www.support-vector.ws (Free educational MATLAB based software for SVMs, NN and FL , Links, Publications downloads, Semisupervised learning software SemiL, Links)
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Software
Lush -- an 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.
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
PCP -- C program for supervised pattern classification. Includes LIBSVM wrapper.
TinySVM -- a small SVM implementation, written in C++
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Interactive SVM applications
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