Closed crowlogic closed 1 month ago
Expression of the Identity: The resolution of the identity can be expressed as:
I = \sum_{k} \phi_k(x) \phi_k^*(y)
where \phi_k are the eigenfunctions (basis functions) of the kernel associated with the RKHS, and I is the identity operator.
Convergence and Function Approximation:
f(x) = \sum_{k} \langle f, \phi_k \rangle \phi_k(x)
In this context, the sums converge in the norm of the RKHS, and as you include more terms from your basis, you get closer to the true function f(x).
Relation to the Kernel:
K(x, y) = \sum_{k} \lambda_k \phi_k(x) \phi_k^*(y)
where \lambda_k are the eigenvalues corresponding to⬤
Resolution of the Identity in RKHS
Expression of the Identity: The resolution of the identity can be expressed as:
I = \sum_{k} \phi_k(x) \phi_k^*(y)
where \phi_k are the eigenfunctions (basis functions) of the kernel associated with the RKHS, and I is the identity operator.
Convergence and Function Approximation:
f(x) = \sum_{k} \langle f, \phi_k \rangle \phi_k(x)
In this context, the sums converge in the norm of the RKHS, and as you include more terms from your basis, you get closer to the true function f(x).
Relation to the Kernel:
K(x, y) = \sum_{k} \lambda_k \phi_k(x) \phi_k^*(y)
where \lambda_k are the eigenvalues corresponding to⬤