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In machine learning, the kernel trick is a method for converting a linear classifier algorithm into a non-linear one by using a non-linear function to map the original observations into a higher-dimensional space; this makes a linear classification in the new space equivalent to non-linear classification in the original space. As a broad subfield of artificial intelligence, Machine learning is concerned with the development of algorithms and techniques that allow computers to learn. At a general level, there are two types of learning: inductive, and deductive. ...
The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...
This is done using Mercer's theorem, which states that any contiuous, symmetric, positive semi-definite kernel function K(x, y) can be expressed as a dot product in a high-dimensional space. In mathematics and functional analysis Mercers theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. ...
In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
Space has been an interest for philosophers and scientists for much of human history. ...
More specifically, if the arguments to the kernel are in a measurable space X, and if the kernel is positive semi-definite — i.e. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...
In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
 for any finite subset {x1, ..., xn} of X and subset {c1, ..., cn} of real numbers — then there exists a function φ(x) whose range is in an inner product space of possibly high dimension, such that In mathematics, the range of a function is the set of all output values produced by that function. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
 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). Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...
The kernel trick was first published by Aizerman et al.[1]
It has been applied to several kinds of algorithm in machine learning and statistics, including: As a broad subfield of artificial intelligence, Machine learning is concerned with the development of algorithms and techniques that allow computers to learn. At a general level, there are two types of learning: inductive, and deductive. ...
A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ...
Who coined the term kernel trick is not known. The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt. ...
Support vector machines (SVMs) are a set of related supervised learning methods used for classification and regression. ...
In statistics, principal components analysis (PCA) is a technique for simplifying a dataset. ...
Sir Ronald Aylmer Fisher, FRS (17 February 1890 – 29 July 1962) was a British statistician, evolutionary biologist, and geneticist. ...
Linear discriminant analysis (LDA) and the related Fishers linear discriminant are used in statistics to find the linear combination of features which best separate two or more classes of object or event. ...
Data clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics. ...
References - ^ M. Aizerman, E. Braverman, and L. Rozonoer (1964). "Theoretical foundations of the potential function method in pattern recognition learning". Automation and Remote Control 25: 821-837.
See also In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ...
In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a function space in which pointwise evaluation is a continuous linear functional. ...
In mathematics and functional analysis Mercers theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. ...
External links - Carlos C. Rodríguez: The Kernel Trick (pdf)
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