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 \(\mathcal{M}\) the latter is given by the formula

\[\begin{split} k_{\nu, \kappa}(x,x') \!=\! \frac{1}{C_{\nu, \kappa}} \sum_{j=0}^{J-1} \Phi_{\nu, \kappa}(\lambda_j) f_j(x) f_j(x') \quad \Phi_{\nu, \kappa}(\lambda) \!=\! \begin{cases} \left(\frac{2\nu}{\kappa^2} + \lambda\right)^{-\nu-\frac{d}{2}} & \nu < \infty \text{ — Matérn} \\ e^{-\frac{\kappa^2}{2} \lambda} & \nu = \infty \text{ — Heat (RBF)} \end{cases} \end{split}\]
The notation here is as follows.

  • The values \(\lambda_j \geq 0\) and the functions \(f_j(\cdot)\) are eigenvalues and eigenfunctions of the minus Laplace–Beltrami operator \(-\Delta_{\mathcal{M}}\) on \(\mathcal{M}\) such that

    \[ \Delta_{\mathcal{M}} f_j = - \lambda_j f_j . \]
    The functions \(\left\{f_j\right\}_{j=0}^{\infty}\) constitute an orthonormal basis of the space \(L^2(\mathcal{M})\) of square integrable functions on the manifold \(\mathcal{M}\) with respect to the inner product \(\langle f, g \rangle_{L^2(\mathcal{M})} = \frac{1}{\lvert\mathcal{M}\rvert} \int_{\mathcal{M}} f(x) g(x) \mathrm{d} x\), where \(\lvert\mathcal{M}\rvert\) denotes the volume of the manifold \(\mathcal{M}\).

  • \(d\) is the dimension of the manifold.

  • The number of eigenpairs \(1 \leq J < \infty\) controls the quality of approximation of the kernel. For some manifolds, e.g. manifolds represented by discrete meshes, this corresponds to the number of levels parameter of the MaternKarhunenLoeveKernel. For others, for which the addition theorem holds (see the respective page), the number of levels parameter has a different meaning [2].

  • \(C_{\nu, \kappa}\) is the constant which ensures that average variance is equal to \(1\), i.e. \(\frac{1}{\lvert\mathcal{M}\rvert}\int_{\mathcal{M}} k(x, x) \mathrm{d} x = 1\). It is easy to show that \(C_{\nu, \kappa} = \sum_{j=0}^{J-1} \Phi_{\nu, \kappa}(\lambda_j)\).

Note: For general manifolds, \(k(x, x)\) can vary from point to point. You usually observe this for manifolds represented by meshes, the ones which do not have a lot of symmetries. On the other hand, for the hyperspheres \(k(x, x)\) is a constant, as it is for all homogeneous spaces which hyperspheres are instances of, as well as for Lie groups (which are also instances of homogeneous spaces).

Footnotes