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Concept
Positive-definite kernel
Résumé
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas.
Let be a nonempty set, sometimes referred to as the index set. A symmetric function is called a positive-definite (p.d.) kernel on if
holds for any , given .
In probability theory, a distinction is sometimes made between positive-definite kernels, for which equality in (1.1) implies , and positive semi-definite (p.s.d.) kernels, which do not impose this condition. Note that this is equivalent to requiring that any finite matrix constructed by pairwise evaluation, , has either entirely positive (p.d.) or nonnegative (p.s.d.) eigenvalues.
In mathematical literature, kernels are usually complex valued functions, but in this article we assume real-valued functions, which is the common practice in applications of p.d. kernels.
For a family of p.d. kernels
The conical sum is p.d., given
The product is p.d., given
The limit is p.d. if the limit exists.
If is a sequence of sets, and a sequence of p.d. kernels, then both and are p.d. kernels on .
Let . Then the restriction of to is also a p.d. kernel.
Common examples of p.d. kernels defined on Euclidean space include:
Linear kernel: .
Polynomial kernel: .
Gaussian kernel (RBF kernel): .
Laplacian kernel: .
Abel kernel: .
Kernel generating Sobolev spaces : , where is the Bessel function of the third kind.
Kernel generating Paley–Wiener space: .
If is a Hilbert space, then its corresponding inner product is a p.
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