Wednesday 17 March 2010

Kernel Functions for Machine Learning Applications

In recent years, Kernel methods have received major attention, particularly due to the increased popularity of the Support Vector Machines. Kernel functions can be used in many applications as they provide a simple bridge from linearity to non-linearity for algorithms which can be expressed in terms of dot products. In this article, we will list a few kernel functions and some of their properties.



Many of these functions have been incorporated in Accord.NET, a extension framework for the popular AForge.NET Framework which also includes many other statistics and machine learning tools.

Contents



  1. Kernel Methods

    1. The Kernel Trick

    2. Kernel Properties

    3. Choosing the Right Kernel



  2. Kernel Functions

    1. Linear Kernel

    2. Polynomial Kernel

    3. Gaussian Kernel

    4. Exponential Kernel

    5. Laplacian Kernel

    6. ANOVA Kernel

    7. Hyperbolic Tangent (Sigmoid) Kernel

    8. Rational Quadratic Kernel

    9. Multiquadric Kernel

    10. Inverse Multiquadric Kernel

    11. Circular Kernel

    12. Spherical Kernel

    13. Wave Kernel

    14. Power Kernel

    15. Log Kernel

    16. Spline Kernel

    17. B-Spline Kernel

    18. Bessel Kernel

    19. Cauchy Kernel

    20. Chi-Square Kernel

    21. Histogram Intersection Kernel

    22. Generalized Histogram Intersection Kernel

    23. Generalized T-Student Kernel

    24. Bayesian Kernel

    25. Wavelet Kernel



  3. Source code

  4. See also

  5. References



 



Kernel Methods



Kernel methods are a class of algorithms for pattern analysis or recognition, whose best known element is the support vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (such as clusters, rankings, principal components, correlations, classifications) in general types of data (such as sequences, text documents, sets of points, vectors, images, graphs, etc) (Wikipedia, 2010a).



The main characteristic of Kernel Methods, however, is their distinct approach to this problem. Kernel methods map the data into higher dimensional spaces in the hope that in this higher-dimensional space the data could become more easily separated or better structured. There are also no constraints on the form of this mapping, which could even lead to infinite-dimensional spaces. This mapping function, however, hardly needs to be computed because of a tool called the kernel trick.



The Kernel Trick


The kernel trick is a mathematical tool which can be applied to any algorithm which solely depends on the dot product between two vectors. Wherever a dot product is used, it is replaced by a kernel function. When properly applied, those candidate linear algorithms are transformed into a non-linear algorithms (sometimes with little effort or reformulation). Those non-linear algorithms are equivalent to their linear originals operating in the range space of a feature space φ. However, because kernels are used, the φ function does not need to be ever explicitly computed. This is highly desirable, as we noted previously, because this higher-dimensional feature space could even be infinite-dimensional and thus infeasible to compute. There are also no constraints on the nature of the input vectors. Dot products could be defined between any kind of structure, such as trees or strings.



Kernel Properties



Kernel functions must be continuous, symmetric, and most preferably should have a positive (semi-) definite Gram matrix. Kernels which are said to satisfy the Mercer's theorem are positive semi-definite, meaning their kernel matrices have no non-negative Eigen values. The use of a positive definite kernel insures that the optimization problem will be convex and solution will be unique.



However, many kernel functions which aren’t strictly positive definite also have been shown to perform very well in practice. An example is the Sigmoid kernel, which, despite its wide use, it is not positive semi-definite for certain values of its parameters. Boughorbel (2005) also experimentally demonstrated that Kernels which are only conditionally positive definite can possibly outperform most classical kernels in some applications.

Kernels also can be classified as anisotropic stationary, isotropic stationary, compactly supported, locally stationary, nonstationary or separable nonstationary. Moreover, kernels can also be labeled scale-invariant or scale-dependant, which is an interesting property as scale-invariant kernels drive the training process invariant to a scaling of the data.



Choosing the Right Kernel

Choosing the most appropriate kernel highly depends on the problem at hand - and fine tuning its parameters can easily become a tedious and cumbersome task. Automatic kernel selection is possible and is discussed in the works by Tom Howley and Michael Madden.

The choice of a Kernel depends on the problem at hand because it depends on what we are trying to model. A polynomial kernel, for example, allows us to model feature conjunctions up to the order of the polynomial. Radial basis functions allows to pick out circles (or hyperspheres) - in constrast with the Linear kernel, which allows only to pick out lines (or hyperplanes).

The motivation behind the choice of a particular kernel can be very intuitive and straightforward depending on what kind of information we are expecting to extract about the data. Please see the final notes on this topic from Introduction to Information Retrieval, by Manning, Raghavan and Schütze for a better explanation on the subject.

 



Kernel Functions

Below is a list of some kernel functions available from the existing literature. As was the case with previous articles, every LaTeX notation for the formulas below are readily available from their alternate text html tag. I can not guarantee all of them are perfectly correct, thus use them at your own risk. Most of them have links to articles where they have been originally used or proposed.



1. Linear Kernel


The Linear kernel is the simplest kernel function. It is given by the inner product <x,y> plus an optional constant c. Kernel algorithms using a linear kernel are often equivalent to their non-kernel counterparts, i.e. KPCA with linear kernel is the same as standard PCA.

k(x, y) = x^T y + c



2. Polynomial Kernel


The Polynomial kernel is a non-stationary kernel. Polynomial kernels are well suited for problems where all the training data is normalized.

k(x, y) = (\alpha x^T y + c)^d  
Adjustable parameters are the slope alpha, the constant term c and the polynomial degree d.


3. Gaussian Kernel

The Gaussian kernel is an example of radial basis function kernel.

k(x, y) = \exp\left(-\frac{ \lVert x-y \rVert ^2}{2\sigma^2}\right)

Alternatively, it could also be implemented using

k(x, y) = \exp\left(- \gamma \lVert x-y \rVert ^2 )

The adjustable parameter sigma plays a major role in the performance of the kernel, and should be carefully tuned to the problem at hand. If overestimated, the exponential will behave almost linearly and the higher-dimensional projection will start to lose its non-linear power. In the other hand, if underestimated, the function will lack regularization and the decision boundary will be highly sensitive to noise in training data.



4. Exponential Kernel

The exponential kernel is closely related to the Gaussian kernel, with only the square of the norm left out. It is also a radial basis function kernel.

k(x, y) = \exp\left(-\frac{ \lVert x-y \rVert }{2\sigma^2}\right)



5. Laplacian Kernel

The Laplace Kernel is completely equivalent to the exponential kernel, except for being less sensitive for changes in the sigma parameter. Being equivalent, it is also a radial basis function kernel.

k(x, y) = \exp\left(- \frac{\lVert x-y \rVert }{\sigma}\right)

It is important to note that the observations made about the sigma parameter for the Gaussian kernel also apply to the Exponential and Laplacian kernels.



6. ANOVA Kernel

The ANOVA kernel is also a radial basis function kernel, just as the Gaussian and Laplacian kernels. It is said to perform well in multidimensional regression problems (Hofmann, 2008).



k(x, y) = \sum_{k=1}^n \exp (-\sigma (x^k - y^k)^2)^d



7. Hyperbolic Tangent (Sigmoid) Kernel

The Hyperbolic Tangent Kernel is also known as the Sigmoid Kernel and as the Multilayer Perceptron (MLP) kernel. The Sigmoid Kernel comes from the Neural Networks field, where the bipolar sigmoid function is often used as an activation function for artificial neurons.

k(x, y) = \tanh (\alpha x^T y + c)

It is interesting to note that a SVM model using a sigmoid kernel function is equivalent to a two-layer, perceptron neural network. This kernel was quite popular for support vector machines due to its origin from neural network theory. Also, despite being only conditionally positive definite, it has been found to perform well in practice.

There are two adjustable parameters in the sigmoid kernel, the slope alpha and the intercept constant c. A common value for alpha is 1/N, where N is the data dimension. A more detailed study on sigmoid kernels can be found in the works by Hsuan-Tien and Chih-Jen.



8. Rational Quadratic Kernel

The Rational Quadratic kernel is less computationally intensive than the Gaussian kernel and can be used as an alternative when using the Gaussian becomes too expensive.

k(x, y) = 1 - \frac{\lVert x-y \rVert^2}{\lVert x-y \rVert^2 + c}



9. Multiquadric Kernel

The Multiquadric kernel can be used in the same situations as the Rational Quadratic kernel. As is the case with the Sigmoid kernel, it is also an example of an non-positive definite kernel.

 k(x, y) = \sqrt{\lVert x-y \rVert^2 + c^2}



10. Inverse Multiquadric Kernel

The Inverse Multi Quadric kernel. As with the Gaussian kernel, it results in a kernel matrix with full rank (Micchelli, 1986) and thus forms a infinite dimension feature space.

k(x, y) = \frac{1}{\sqrt{\lVert x-y \rVert^2 + \theta^2}}



11. Circular Kernel


The circular kernel is used in geostatic applications. It is an example of an isotropic stationary kernel and is positive definite in R2.


k(x, y) = \frac{2}{\pi} \arccos ( - \frac{ \lVert x-y \rVert}{\sigma}) - \frac{2}{\pi} \frac{ \lVert x-y \rVert}{\sigma} \sqrt{1 - \left(\frac{ \lVert x-y \rVert}{\sigma} \right)^2}
\mbox{if}~ \lVert x-y \rVert < \sigma \mbox{, zero otherwise}



12. Spherical Kernel

The spherical kernel is similar to the circular kernel, but is positive definite in R3.

k(x, y) = 1 - \frac{3}{2} \frac{\lVert x-y \rVert}{\sigma} + \frac{1}{2} \left( \frac{ \lVert x-y \rVert}{\sigma} \right)^3

\mbox{if}~ \lVert x-y \rVert < \sigma \mbox{, zero otherwise}



13. Wave Kernel

The Wave kernel is also symmetric positive semi-definite (Huang, 2008).

k(x, y) = \frac{\theta}{\lVert x-y \rVert \right} \sin \frac{\lVert x-y \rVert }{\theta}



14. Power Kernel

The Power kernel is also known as the (unrectified) triangular kernel. It is an example of scale-invariant kernel (Sahbi and Fleuret, 2004) and is also only conditionally positive definite.

k(x,y) = - \lVert x-y \rVert ^d



15. Log Kernel

The Log kernel seems to be particularly interesting for images, but is only conditionally positive definite.

k(x,y) = - log (\lVert x-y \rVert ^d + 1)



16. Spline Kernel

The Spline kernel is given as a piece-wise cubic polynomial, as derived in the works by Gunn (1998).

k(x, y) = 1 + xy + xy~min(x,y) - \frac{x+y}{2}~min(x,y)^2+\frac{1}{3}\min(x,y)^3

However, what it actually mean is:

k(x,y) = \prod_{i=1}^d 1 + x_i y_i + x_i y_i \min(x_i, y_i) - \frac{x_i + y_i}{2} \min(x_i,y_i)^2 + \frac{\min(x_i,y_i)^3}{3}

Withx,y \in R^d

 



17. B-Spline (Radial Basis Function) Kernel

The B-Spline kernel is defined on the interval [−1, 1]. It is given by the recursive formula:

k(x,y) = B_{2p+1}(x-y)

\mbox{where~} p \in N \mbox{~with~} B_{i+1} := B_i \otimes B_0.

 

In the work by Bart Hamers it is given by:k(x, y) = \prod_{p=1}^d B_{2n+1}(x_p - y_p)

 

Alternatively, Bn can be computed using the explicit expression (Fomel, 2000):

B_n(x) = \frac{1}{n!} \sum_{k=0}^{n+1} \binom{n+1}{k} (-1)^k (x + \frac{n+1}{2} - k)^n_+

 

Where x+ is defined as the truncated power function:

 x^d_+ = \begin{cases} x^d, & \mbox{if }x > 0 \\ 0, & \mbox{otherwise} \end{cases}



18. Bessel Kernel

The Bessel kernel is well known in the theory of function spaces of fractional smoothness. It is given by:

k(x, y) = \frac{J_{v+1}( \sigma \lVert x-y \rVert)}{ \lVert x-y \rVert ^ {-n(v+1)} }

where J is the Bessel function of first kind. However, in the Kernlab for R documentation, the Bessel kernel is said to be:

k(x,x') = - Bessel_{(nu+1)}^n (\sigma |x - x'|^2)



19. Cauchy Kernel

The Cauchy kernel comes from the Cauchy distribution (Basak, 2008). It is a long-tailed kernel and can be used to give long-range influence and sensitivity over the high dimension space.

k(x, y) = \frac{1}{1 + \frac{\lVert x-y \rVert^2}{\sigma^2} }



20. Chi-Square Kernel

The Chi-Square kernel comes from the Chi-Square distribution.

k(x,y) = 1 - \sum_{i=1}^n \frac{(x_i-y_i)^2}{\frac{1}{2}(x_i+y_i)}



21. Histogram Intersection Kernel

The Histogram Intersection Kernel is also known as the Min Kernel and has been proven useful in image classification.

k(x,y) = \sum_{i=1}^n \min(x_i,y_i)



22. Generalized Histogram Intersection

The Generalized Histogram Intersection kernel is built based on the Histogram Intersection Kernel for image classification but applies in a much larger variety of contexts (Boughorbel, 2005). It is given by:

k(x,y) = \sum_{i=1}^m \min(|x_i|^\alpha,|y_i|^\beta)



23. Generalized T-Student Kernel

The Generalized T-Student Kernel has been proven to be a Mercel Kernel, thus having a positive semi-definite Kernel matrix (Boughorbel, 2004). It is given by:

k(x,y) = \frac{1}{1 + \lVert x-y \rVert ^d}



24. Bayesian Kernel

The Bayesian kernel could be given as:

k(x,y) = \prod_{l=1}^N \kappa_l (x_l,y_l)

where

\kappa_l(a,b) = \sum_{c \in \{0;1\}} P(Y=c \mid X_l=a) ~ P(Y=c \mid X_l=b)

However, it really depends on the problem being modeled. For more information, please see the work by Alashwal, Deris and Othman, in which they used a SVM with Bayesian kernels in the prediction of protein-protein interactions.



25. Wavelet Kernel


The Wavelet kernel (Zhang et al, 2004) comes from Wavelet theory and is given as:



k(x,y) = \prod_{i=1}^N h(\frac{x_i-c_i}{a}) \: h(\frac{y_i-c_i}{a})

Where a and c are the wavelet dilation and translation coefficients, respectively (the form presented above is a simplification, please see the original paper for details). A translation-invariant version of this kernel can be given as:



k(x,y) = \prod_{i=1}^N h(\frac{x_i-y_i}{a})

Where in both h(x) denotes a mother wavelet function. In the paper by Li Zhang, Weida Zhou, and Licheng Jiao, the authors suggests a possible h(x) as:



h(x) = cos(1.75x)exp(-\frac{x^2}{2})

Which they also prove as an admissible kernel function.



 



Source Code


The latest version of the source code for almost all of the kernels listed above is available in the Accord.NET Framework. Some are also available in the sequel of this article, Kernel Support Vector Machines for Classification and Regression in C#. They are provided together with a comprehensive and simple implementation of SVMs (Support Vector Machines) in C#. However, for the latest sources, which may contain bug fixes and other enhancements, please download the most recent version available of Accord.NET.



 


See also




 

References



Citing this work


If you would like, please cite this work as: Souza, César R. "Kernel Functions for Machine Learning Applications." 17 Mar. 2010. Web. <http://crsouza.blogspot.com/2010/03/kernel-functions-for-machine-learning.html>.

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