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hermitian<T,N>

Description

Hermitian<T,N> is a family of Kernels. Specifically, it is a family of Mercer Kernels. The N-th Hermitian kernel is equal to the N-th order derivative of the Gaussian kernel. They are named after Hermite polynomials, which may be used to identify the derivatives of the Gaussian kernels.

If k is an object of class hermite<T,N>, u and v are objects of class T, and N is a integral constant, then k(u,v) returns

(1)
eqn_8xz2

where eqn_v66o is Rodrigues’ formula for Hermite polynomials

(2)
eqn_khx7

Figure 1 shows the Hermitian kernel for N=0 through N=5 for a scalar input type.

eqn_gbu6

Figure 1: Hermitian kernels, parametrised by eqn_12ny, with N=0 at the top left through N=5 at the bottom right.

Example

vector< double > x(10); 
vector< double > v(10); 
hermitian< vector< double >, 2 > kernel(1.0); 
cout << kernel( x, v ) << endl; 

Definition

Defined in the KML header <kml/hermitian.hpp>.

Template Parameters

Parameter Description Default
T The hermitian argument type
N The order of the hermitian

Model of

Mercer Kernel

Type requirements

T must be a vector type or a scalar type.

Members

Member Where defined Description
hermitian() Default Constructible The default constructor
result_type Input value The type of the result: input_value<T>

Notes

See also

Mercer Kernel, linear, gaussian, polynomial, sigmoid