Kernels on Compact Manifolds¶
Warning
You can get by fine without reading this page for almost all use cases, just use the standard MaternGeometricKernel
, following the example notebook on hypersheres.
This is optional material meant to explain the basic theory and based mainly on Borovitskiy et al. [2020]. [1]
Theory¶
For compact Riemannian manifolds, MaternGeometricKernel
is an alias to MaternKarhunenLoeveKernel
.
For such a manifold
The values
and the functions are eigenvalues and eigenfunctions of the minus Laplace–Beltrami operator on such that constitute an orthonormal basis of the space of square integrable functions on the manifold with respect to the inner product , where denotes the volume of the manifold . is the dimension of the manifold.The number of eigenpairs
controls the quality of approximation of the kernel. For some manifolds, e.g. manifolds represented by discretemeshes
, this corresponds to the number of levels parameter of theMaternKarhunenLoeveKernel
. For others, for which the addition theorem holds (see the respective page), the number of levels parameter has a different meaning [2]. is the constant which ensures that average variance is equal to , i.e. . It is easy to show that .
Note: For general manifolds,
Footnotes