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In machine learning, the kernel trick is a method for using a linear classifier algorithm to solve a non-linear problem by mapping the original non-linear observations into a higher-dimensional space, where the linear classifier is subsequently used; this makes a linear classification in the new space equivalent to non-linear classification in the original space.## References

## See also

This is done using Mercer's theorem, which states that any continuous, symmetric, positive semi-definite kernel function K(x, y) can be expressed as a dot product in a high-dimensional space.

More specifically, if the arguments to the kernel are in a measurable space X, and if the kernel is positive semi-definite — i.e.

- $sum\_\{i,j\}\; K(x\_i,x\_j)\; c\_i\; c\_j\; ge\; 0$

for any finite subset {x_{1}, ..., x_{n}} of X and subset {c_{1}, ..., c_{n}} of objects (typically real numbers) — then there exists a function φ(x) whose range is in an inner product space of possibly high dimension, such that

- $K(x,y)\; =\; varphi(x)cdotvarphi(y).$

The kernel trick transforms any algorithm that solely depends on the dot product between two vectors. Wherever a dot product is used, it is replaced with the kernel function. Thus, a linear algorithm can easily be transformed into a non-linear algorithm. This non-linear algorithm is equivalent to the linear algorithm operating in the range space of φ. However, because kernels are used, the φ function is never explicitly computed. This is desirable, because the high-dimensional space may be infinite-dimensional (as is the case when the kernel is a Gaussian).

The kernel trick was first published by Aizerman et al.

It has been applied to several kinds of algorithm in machine learning and statistics, including:

- Perceptrons
- Support vector machines
- Principal components analysis
- Canonical correlation analysis
- Fisher's linear discriminant analysis
- Clustering

The origin of the term kernel trick is not known.

- Kernel methods
- Integral transforms
- Hilbert space, specifically reproducing kernel Hilbert space
- Mercer kernel

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Last updated on Thursday July 10, 2008 at 23:42:55 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday July 10, 2008 at 23:42:55 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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