Addition Theorem

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].

Theory

This builds on the general theory on compact manifolds and uses the same notation.

Consider a hypersphere: \(M = \mathbb{S}_d\). Then closed form expressions for \(\lambda_j\) and \(f_j\) are known (see, e.g., Appendix B of Borovitskiy et al. [2020]). The eigenfunctions \(f_j\) in this case are the (hyper)spherical harmonics, restrictions of certain known polynomials in \(\mathbb{R}^{d+1}\) on the unit sphere.

However, although \(\lambda_j, f_j\) are known for \(M = \mathbb{S}_d\), using the general formula for \(k(x, x')\) from the compact manifolds page is suboptimal in this case. This is so because of the following theorem.

Addition theorem. The spherical harmonics \(f_j\) can be re-indexed as \(f_{l s}\) with \(l = 0, \ldots, \infty\) and \(s = 1, \ldots, d_l\) with \(d_l = (2l+d-1) \frac{\Gamma(l+d-1)}{\Gamma(d) \Gamma(l+1)}\) [1] such that

  • all eigenfunctions in the set \(\{f_{l s}\}_{s=1}^{d_l}\) correspond to the same eigenvalue \(\lambda_l = l(l+d-1)\),

  • the following equation holds

    \[ \sum_{s=1}^{d_l} f_{l s}(x) f_{l s}(x') = c_{l, d} \mathcal{C}^{(d-1)/2}_l(\cos(\mathrm{d}_{\mathbb{S}_d}(x, x'))) \qquad c_{l, d} = d_l \frac{\Gamma((d+1)/2)}{2 \pi^{(d+1)/2} \mathcal{C}_l^{(d-1)/2}(1)} , \]
    where \(\mathcal{C}^{(d-1)/2}_l\) are certain known polynomials called Gegenbauer polynomials and \(\mathrm{d}_{\mathbb{S}_d}\) is the geodesic distance on the (hyper)sphere.

Thanks to this, we have

\[\begin{split} k_{\nu, \kappa}(x,x') = \frac{1}{C_{\nu, \kappa}} \sum_{l=0}^{L-1} \Phi_{\nu, \kappa}(\lambda_l) c_{l, d} \mathcal{C}^{(d-1)/2}_l(\cos(\mathrm{d}_{\mathbb{S}_d}(x, x'))) \qquad \Phi_{\nu, \kappa}(\lambda) = \begin{cases} \left(\frac{2\nu}{\kappa^2} + \lambda\right)^{-\nu-\frac{d}{2}} & \nu < \infty \text{ — Matérn} \\ e^{-\frac{\kappa^2}{2} \lambda} & \nu = \infty \text{ — Heat (RBF)} \end{cases} \end{split}\]
which is more efficient to use than the general formula above. The reason is simple: it is not harder to evaluate a Gegenbauer polynomial \(\mathcal{C}^{(d-1)/2}_l\) than each single one of the respective (hyper)spherical harmonics. At the same time, you need much fewer Gegenbauer polynomials to achieve the same quality of approximation. For example, for \(M = \mathbb{S}_2\) and \(L = 20\) the corresponding \(J\) is \(400\).

Note

The \(l\) in the example above indexes what we call levels in the library. These are certain sets of eigenfunctions that correspond to the same eigenvalue (not necessarily a maximal set of those, i.e. not necessarily the full eigenspace), for which one can efficiently compute the outer product \(\sum_{s} f_{l s}(x) f_{l s}(x')\) without having to compute the individual eigenfunctions. [2]

Note

In the simplest special case of \(\mathbb{S}_d\), the circle \(\mathbb{S}_1\), the eigenfunctions are given by \(\sin(l \theta), \cos(l \theta)\), where \(l\) indexes levels. The outer product \(\cos(l \theta) \cos(l \theta') + \sin(l \theta) \sin(l \theta')\) in this case can be simplified to \(\cos(l (\theta-\theta')) = \cos(l d_{\mathbb{S}_1}(\theta, \theta'))\) thanks to an elementary trigonometric identity.

Such addition theorems appear beyond hyperspheres, for example for Lie groups and other compact homogeneous spaces [Azangulov et al., 2024]. In the library, such spaces use the class EigenfunctionsWithAdditionTheorem to represent the spectrum of \(\Delta_{\mathcal{M}}\). For them, the number of levels parameter of the MaternKarhunenLoeveKernel maps to \(L\) in the above formula.

Footnotes