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Mercel Kernel

Description

A Mercer Kernel is a Kernel that is positive semi-definite. When a kernel is positive semi-definite, one may exploit the kernel trick [1], the idea of mapping data to a high-dimensional feature space where some linear algorithm is applied that works exclusively with inner products. Assume we have some mapping eqn_p0m9 from an input space eqn_0j1j to a feature space eqn_jwxw, then a kernel function (or kernel)

eqn_zftg

may be used to define the inner product in feature space eqn_jwxw. Figure 1 shows that the application of a linear algorithm in feature space eqn_jwxw could correspond to a nonlinear estimate in input space eqn_0j1j.

feature_space

Figure 1: A mapping eqn_p0m9 from input space eqn_0j1j to feature space eqn_jwxw.

Positive definiteness in the context of kernel functions also implies that a kernel matrix created using a particular kernel is positive semi-definite. A matrix is positive semi-definite if its associated eigenvalues eqn_3msm are nonnegative.

Refinement of

Kernel

Associated types

Notation

Definitions

Valid Expressions

Expression Semantics

Complexity Guarantees

Invariants

Models

Notes

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References

[1]

Mark Aizerman, Emmanuil Braverman, and Lev Rozonoèr. Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control, 25:821–837, 1964.