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Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time series , in the use of the periodogram to estimate the spectral density where they are known as window functions .
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 ...
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. [ 1 ]
The kernel is a subrng, and, more precisely, a two-sided ideal of R. Thus, it makes sense to speak of the quotient ring R / (ker f). The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f (which is a subring of S). (Note that rings need not be unital for the kernel definition).
Kernel (linear algebra) or null space, a set of vectors mapped to the zero vector; Kernel (category theory), a generalization of the kernel of a homomorphism; Kernel (set theory), an equivalence relation: partition by image under a function; Difference kernel, a binary equalizer: the kernel of the difference of two functions
The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel.
The term graph kernels was more officially coined in 2002 by R. I. Kondor and J. Lafferty [4] as kernels on graphs, i.e. similarity functions between the nodes of a single graph, with the World Wide Web hyperlink graph as a suggested application. In 2003, Gärtner et al. [5] and Kashima et al. [6] defined kernels between graphs.
The kernel of the empty set, , is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty. [3] A family is said to be free if it is not fixed; that is, if its kernel is the empty set. [3]