Support vector machine

Support vector machines (SVMs) are a set of related supervised learning methods used for classification and regression. A support vector machine constructs a hyperplane or set of hyperplanes in a high or infinite dimensional space, which can be used for classification, regression or other tasks. Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training datapoints of any class (so-called functional margin), since in general the larger the margin the lower the generalization error of the classifier.

Motivation

H3 (green) doesn't separate the 2 classes. H1 (blue) does, with a small margin and H2 (red) with the maximum margin.

Classifying data is a common task in machine learning. Suppose some given data points each belong to one of two classes, and the goal is to decide which class a new data point will be in. In the case of support vector machines, a data point is viewed as a $p$-dimensional vector (a list of $p$ numbers), and we want to know whether we can separate such points with a $p-1$-dimensional hyperplane. This is called a linear classifier. There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the maximum-margin hyperplane and the linear classifier it defines is known as a maximum margin classifier.

Formalization

We are given some training data, a set of points of the form

${\mathcal {D}}=\{(\mathbf {x} _{i},c_{i})|\mathbf {x} _{i}\in \mathbb {R} ^{p},c_{i}\in \{-1,1\}\}_{i=1}^{n}$

where the c_i is either 1 or −1, indicating the class to which the point $\mathbf {x} _{i}$ belongs. Each $\mathbf {x} _{i}$ is a $p$-dimensional real vector. We want to find the maximum-margin hyperplane which divides the points having $c_{i}=1$ from those having $c_{i}=-1$. Any hyperplane can be written as the set of points $\mathbf {x}$ satisfying

Maximum-margin hyperplane and margins for a SVM trained with samples from two classes. Samples on the margin are called the support vectors.

$\mathbf {w} \cdot \mathbf {x} -b=0,\,$

where $\cdot$ denotes the dot product. The vector ${\mathbf {w} }$ is a normal vector: it is perpendicular to the hyperplane. The parameter ${\tfrac {b}{\|\mathbf {w} \|}}$ determines the offset of the hyperplane from the origin along the normal vector ${\mathbf {w} }$.

We want to choose the ${\mathbf {w} }$ and $b$ to maximize the margin, or distance between the parallel hyperplanes that are as far apart as possible while still separating the data. These hyperplanes can be described by the equations

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

and

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

Note that if the training data are linearly separable, we can select the two hyperplanes of the margin in a way that there are no points between them and then try to maximize their distance. By using geometry, we find the distance between these two hyperplanes is ${\tfrac {2}{\|\mathbf {w} \|}}$, so we want to minimize $\|\mathbf {w} \|$. As we also have to prevent data points falling into the margin, we add the following constraint: for each $i$ either

$\mathbf {w} \cdot \mathbf {x} _{i}-b\geq 1\qquad {\text{ for }}\mathbf {x} _{i}$

of the first class or

$\mathbf {w} \cdot \mathbf {x} _{i}-b\leq -1\qquad {\text{ for }}\mathbf {x} _{i}$ of the second.

This can be rewritten as:

$c_{i}(\mathbf {w} \cdot \mathbf {x} _{i}-b)\geq 1,\quad {\text{ for all }}1\leq i\leq n.\qquad \qquad (1)$

We can put this together to get the optimization problem:

Minimize (in ${\mathbf {w} ,b}$)

$\|\mathbf {w} \|\,$

subject to (for any $i=1,\dots ,n$)

$c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)\geq 1.\,$

Primal form

The optimization problem presented in the preceding section is difficult to solve because it depends on ||w||, the norm of w, which involves a square root. Fortunately it is possible to alter the equation by substituting ||w|| with ${\tfrac {1}{2}}\|\mathbf {w} \|^{2}$ without changing the solution (the minimum of the original and the modified equation have the same w and b). This is a quadratic programming (QP) optimization problem. More clearly:

Minimize (in ${\mathbf {w} ,b}$)

${\frac {1}{2}}\|\mathbf {w} \|^{2}$

subject to (for any $i=1,\dots ,n$)

$c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)\geq 1.$

which can be expressed by means of non-negative Lagrange multipliers $\alpha _{i}$ as

$\min _{\mathbf {w} ,b}\{{\frac {1}{2}}\|\mathbf {w} \|^{2}-\sum _{i=0}^{n}{\alpha _{i}[c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)-1]}\}$

The factor of 1/2 is used for mathematical convenience. This problem can now be solved by standard quadratic programming techniques and programs. The solution can be expressed by terms of linear combination of the training vectors as

$\mathbf {w} =\sum _{i=0}^{n}{\alpha _{i}c_{i}\mathbf {x_{i}} }$

Only few $\alpha _{i}$ will be greater than zero. These are exactly the support vectors $\mathbf {x_{i}}$, which lie on the margin and satisfy $c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)=1$. From this one can derive, that the support vectors also satisfy

$\mathbf {w} \cdot \mathbf {x_{i}} -b=c_{i}\iff b=\mathbf {w} \cdot \mathbf {x_{i}} -c_{i}$

which allows to define the offset $b$. In practice, it is more robust to average over all $N_{SV}$ support vectors:

$b={\frac {1}{N_{SV}}}\sum _{i=0}^{N_{SV}}{(\mathbf {w} \cdot \mathbf {x_{i}} -c_{i})}$

Dual form

Writing the classification rule in its unconstrained dual form reveals that the maximum margin hyperplane and therefore the classification task is only a function of the support vectors, the training data that lie on the margin.

Using the fact, that $\|\mathbf {w} \|^{2}=w\cdot w$ and substituting $\mathbf {w} =\sum _{i=0}^{n}{\alpha _{i}c_{i}\mathbf {x_{i}} }$, one can show that the dual of the SVM boils down to the following optimization problem:

Maximize (in $\alpha _{i}$ )

$\sum _{i=1}^{n}\alpha _{i}-{\frac {1}{2}}\sum _{i,j}\alpha _{i}\alpha _{j}c_{i}c_{j}\mathbf {x} _{i}^{T}\mathbf {x} _{j}$

subject to (for any $i=1,\dots ,n$)

$\alpha _{i}\geq 0,\,$

and

$\sum _{i=1}^{n}\alpha _{i}c_{i}=0.$

The $\alpha$ terms constitute a dual representation for the weight vector in terms of the training set:

$\mathbf {w} =\sum _{i}\alpha _{i}c_{i}\mathbf {x} _{i}.$

Biased and unbiased hyperplanes

For simplicity reasons, sometimes it is required that the hyperplane passes through the origin of the coordinate system. Such hyperplanes are called unbiased, whereas general hyperplanes not necessarily passing through the origin are called biased. An unbiased hyperplane can be enforced by setting $b=0$ in the primal optimization problem. The corresponding dual is identical to the dual given above without the equality constraint

$\sum _{i=1}^{n}\alpha _{i}c_{i}=0.$

Transductive support vector machines

Transductive support vector machines extend SVMs in that they also take into account structural properties (e.g. correlational structures) of the data set to be classified. Here, in addition to the training set ${\mathcal {D}}$, the learner is also given a set

${\mathcal {D}}^{\star }=\{\mathbf {x} _{i}^{\star }|\mathbf {x} _{i}^{\star }\in \mathbb {R} ^{p}\}_{i=1}^{k}\,$

of test examples to be classified. Formally, a transductive support vector machine is defined by the following primal optimization problem:

Minimize (in ${\mathbf {w} ,b,\mathbf {c^{\star }} }$)

${\frac {1}{2}}\|\mathbf {w} \|^{2}$

subject to (for any $i=1,\dots ,n$ and any $j=1,\dots ,k$)

$c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)\geq 1,\,$

$c_{j}^{\star }(\mathbf {w} \cdot \mathbf {x_{j}^{\star }} -b)\geq 1,$

and

$c_{j}^{\star }\in \{-1,1\}.\,$

Transductive support vector machines have been introduced by Vladimir Vapnik in 1998.

Properties

SVMs belong to a family of generalized linear classifiers. They can also be considered a special case of Tikhonov regularization. A special property is that they simultaneously minimize the empirical classification error and maximize the geometric margin; hence they are also known as maximum margin classifiers.

A comparison of the SVM to other classifiers has been made by Meyer, Leisch and Hornik.

Extensions to the linear SVM

Soft margin

In 1995, Corinna Cortes and Vladimir 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. The method introduces slack variables, $\xi _{i}$, which measure the degree of misclassification of the datum $x_{i}$

$c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)\geq 1-\xi _{i}\quad 1\leq i\leq n.\quad \quad (2)$

The objective function is then increased by a function which penalizes non-zero $\xi _{i}$, and the optimization becomes a trade off between a large margin, and a small error penalty. If the penalty function is linear, the optimization problem becomes:

$\operatorname {argmin} {\frac {1}{2}}\|\mathbf {w} \|^{2}+C\sum _{i}\xi _{i}\quad \quad {\text{subject to }}c_{i}(\mathbf {w} \cdot \mathbf {x_{i}} -b)\geq 1-\xi _{i}\quad 1\leq i\leq n.$

This constraint in (2) along with the objective of minimizing $\|\mathbf {w} \|$ 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. For the above formulation and its huge impact in practice, Cortes and Vapnik received the 2008 ACM Paris Kanellakis Award . 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.

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 et al.. ) 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 a 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): $k(\mathbf {x_{i}} ,\mathbf {x_{j}} )=(\mathbf {x_{i}} \cdot \mathbf {x_{j}} )^{d}$
• Polynomial (inhomogeneous): $k(\mathbf {x_{i}} ,\mathbf {x_{j}} )=(\mathbf {x_{i}} \cdot \mathbf {x_{j}} +1)^{d}$
• Radial Basis Function: $k(\mathbf {x_{i}} ,\mathbf {x_{j}} )=\exp(-\gamma \|\mathbf {x_{i}} -\mathbf {x_{j}} \|^{2})$, for $\gamma >0$
• Gaussian Radial basis function: $k(\mathbf {x_{i}} ,\mathbf {x_{j}} )=\exp \left(-{\frac {\|\mathbf {x_{i}} -\mathbf {x_{j}} \|^{2}}{2\sigma ^{2}}}\right)$
• Hyperbolic tangent: $k(\mathbf {x_{i}} ,\mathbf {x_{j}} )=\tanh(\kappa \mathbf {x_{i}} \cdot \mathbf {x_{j}} +c)$, for some (not every) $\kappa >0$ and $c<0$

The kernel is related to the transform $\varphi (\mathbf {x_{i}} )$ by the equation $k(\mathbf {x_{i}} ,\mathbf {x_{j}} )=\varphi (\mathbf {x_{i}} )\cdot \varphi (\mathbf {x_{j}} )$. The value w is also in the transformed space, with $\textstyle \mathbf {w} =\sum _{i}\alpha _{i}c_{i}\varphi (\mathbf {x} _{i}).$ Dot products with w for classification can again be computed by the kernel trick, i.e. $\textstyle \mathbf {w} \cdot \varphi (\mathbf {x} )=\sum _{i}\alpha _{i}c_{i}k(\mathbf {x} _{i},\mathbf {x} )$. However, there does not in general exist a value w' such that $\mathbf {w} \cdot \varphi (\mathbf {x} )=k(\mathbf {w'} ,\mathbf {x} ).$

Issues

Potential drawbacks of the SVM are the following two aspects:

• Uncalibrated Class membership probabilities
• The SVM is only directly applicable for two-class tasks. Therefore, algorithms that reduce the multi-class task to several binary problems have to be applied, see the Multi-class SVM section.

Multiclass SVM

Multiclass SVM aims to assign labels to instances by using support vector machines, where the labels are drawn from a finite set of several elements. The dominating approach for doing so is to reduce the single multiclass problem into multiple binary problems. Each of the problems yields a binary classifier, which is assumed to produce an output function that gives relatively large values for examples from the positive class and relatively small values for examples belonging to the negative class. Two common methods to build such binary classifiers are where each classifier distinguishes between (i) one of the labels to the rest (one-versus-all) or (ii) between every pair of classes (one-versus-one). Classification of new instances for one-versus-all case is done by a winner-takes-all strategy, in which the classifier with the highest output function assigns the class (it is important that the output functions be calibrated to produce comparable scores). For the one-versus-one approach, classification is done by a max-wins voting strategy, in which every classifier assigns the instance to one of the two classes, then the vote for the assigned class is increased by one vote, and finally the class with most votes determines the instance classification.

Structured SVM

SVMs have been generalized to Structured SVM, where the label space is structured and of possibly infinite size.

Regression

A version of SVM for regression was proposed in 1996 by Vladimir 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) depends only 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 depends only on a subset of the training data, because the cost function for building the model ignores any training data close to the model prediction (within a threshold $\epsilon$).

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-manageable chunks. A common method for solving the QP problem is the Platt's Sequential Minimal Optimization (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.

Another approach is to use an interior point method that uses Newton-like iterations to find a solution of the Karush-Kuhn-Tucker conditions of the primal and dual problems. Instead of solving a sequence of broken down problems, this approach directly solves the problem as a whole. To avoid solving a linear system involving the large kernel matrix, a row rank approximation to the matrix is often used to use the kernel trick.

See also

• Kernel machines
• Predictive analytics
• Relevance vector machine, a probabilistic sparse kernel model identical in functional form to SVM.

External links

General

• A Tutorial on Support Vector Machines for Pattern Recognition by Christopher J. C. Burges. Data Mining and Knowledge Discovery 2:121–167, 1998
• www.kernel-machines.org (general information and collection of research papers)
• www.support-vector-machines.org (Literature, Review, Software, Links related to Support Vector Machines — Academic Site)
• videolectures.net (SVM-related video lectures)
• Animation clip: SVM with polynomial kernel visualization.
• A very basic SVM tutorial for complete beginners by Tristan Fletcher [1].

Software

• ADAPA — a batch and real-time PMML based scoring engine for data mining models including Support Vector Machines.
• Algorithm::SVM — Perl bindings for the libsvm Support Vector Machine library
• dlib C++ Library — A C++ library that includes an easy to use SVM classifier
• e1071 — Machine learning library for R
• Gist — implementation of the SVM algorithm with feature selection.
• kernlab — Kernel-based Machine Learning library for R
• LIBLINEAR — A Library for Large Linear Classification, Machine Learning Group at National Taiwan University
• LIBSVM — A Library for Support Vector Machines, Chih-Chung Chang and Chih-Jen Lin
• 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.
• LS-SVMLab — Matlab/C SVM toolbox — well-documented, many features
• mlpy — Machine Learning Py — high-performance Python package for predictive modeling
• NSvm Open source SVM implementation for .NET in C#.
• Oracle Data Mining Option provides 12 in-database data mining functions inside the Oracle Database Enterprise Edition
• OSU SVM — Matlab implementation based on LIBSVM
• Orange, a free data mining software suite, module orngSVM
• pcSVM is an object oriented SVM framework written in C++ and provides wrapping to Python classes. The site provides a stand alone demo tool for experimenting with SVMs.
• PCP — C program for supervised pattern classification. Includes LIBSVM wrapper.
• PyML — a Python machine learning package. Includes: SVM, nearest neighbor classifiers, ridge regression, Multi-class methods (one-against-one and one-against-rest), Feature selection (filter methods, RFE, multiplicative update, Model selection, Classifier testing (cross-validation, error rates, ROC curves, statistical test for comparing classifiers).
• Shogun — Large Scale Machine Learning Toolbox that provides several SVM implementations (like libSVM, SVMlight) under a common framework and interfaces to Octave, Matlab, Python, R
• SimpleSVM — SimpleSVM toolbox for Matlab
• SVM-KM — SVM and Kernel Methodes toolbox for Matlab
• SimpleMKL — Multiple Kernel Learning toolbox for Matlab
• Spider — Machine learning library for Matlab
• Statistical Pattern Recognition Toolbox for Matlab.
• SVM and Kernel Methods Matlab Toolbox
• SVM Classification Applet — Performs classification on any given data set and gives 10-fold cross-validation error rate
• SVMlight — a popular implementation of the SVM algorithm by Thorsten Joachims; it can be used to solve classification, regression and ranking problems.
  • automation of SVMlight in Matlab — complete automation of SVMlight for use with fMRI data
• SVMProt — Protein Functional Family Prediction.
• Torch — C++ machine learning library with SVM
• The Kernel-Machine Library (GNU) C++ template library for Support Vector Machines
• TinySVM — a small SVM implementation, written in C++
• YALE (now RapidMiner) — a powerful machine learning toolbox containing wrappers for SVMLight, LibSVM, and MySVM in addition to many evaluation and preprocessing methods.
• 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. (The SVM implementation is called "SMO". It can be found in the Weka Explorer GUI, under the "functions" category or in the Weka Explorer GUI as SVMAttributeEval, under Select attributes, attributeSelection.)

Interactive SVM applications

• ECLAT classification of Expressed Sequence Tag (EST) from mixed EST pools using codon usage
• EST3 classification of Expressed Sequence Tag (EST) from mixed EST pools using nucleotide triples

Bibliography

• 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 ([2] SVM Book)
• 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[3]
• Vojislav Kecman: "Learning and Soft Computing — Support Vector Machines, Neural Networks, Fuzzy Logic Systems", The MIT Press, Cambridge, MA, 2001.[4]
• Bernhard Schölkopf and A. J. Smola: Learning with Kernels. MIT Press, Cambridge, MA, 2002. (Partly available on line: [5].) 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. [6]
• John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. ISBN 0-521-81397-2 ([7] Kernel Methods Book)
• Ingo Steinwart and Andreas Christmann. Support Vector Machines. Springer-Verlag, New York, 2008. ISBN 978-0-387-77241-7 ([8] SVM 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. (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, 1995. ISBN 0-387-98780-0
• Vladimir Vapnik, S.Kotz "Estimation of Dependences Based on Empirical Data" Springer, 2006. ISBN 0387308652, 510 pages [this is a reprint of Vapnik's early book describing philosophy behind SVM approach. The 2006 Appendix describes recent development].
• Dmitriy Fradkin and Ilya Muchnik "Support Vector Machines for Classification" in J. Abello and G. Carmode (Eds) "Discrete Methods in Epidemiology", DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 70, pp. 13–20, 2006. [9]. Succinctly describes theoretical ideas behind SVM.
• Kristin P. Bennett and Colin Campbell, "Support Vector Machines: Hype or Hallelujah?", SIGKDD Explorations, 2,2, 2000, 1-13. [10]. Excellent introduction to SVMs with helpful figures.
• Ovidiu Ivanciuc, "Applications of Support Vector Machines in Chemistry", In: Reviews in Computational Chemistry, Volume 23, 2007, pp. 291–400. Reprint available: [11]
• Catanzaro, Sundaram, Keutzer, "Fast Support Vector Machine Training and Classification on Graphics Processors", In: International Conference on Machine Learning, 20080 [12]