kν(x,x′)=σ2Γ(ν)21−ν(2νκ∥x−x′∥)νKν(2νκ∥x−x′∥)k∞(x,x′)=σ2exp(−2κ2∥x−x′∥2)
σ2: variance
κ: length scale
ν: smoothness
ν→∞: recovers squared exponential kernel
ν=1/2
ν=3/2
ν=5/2
ν=∞
f:G→R
f()→R
f()→R
f()→R
Mateˊrn(κ22ν−Δ)2ν+4df=Wsquared exponential(κ22ν−Δ)2ν+4de−4κ2Δf=W
Δ: Laplacian
W: (rescaled) white noise
e−4κ2Δ: (rescaled) heat semigroup
(Δf)(x)=x′∼x∑wxx′(f(x)−f(x′)) Δ=D−W D: degree matrix W: (weighted) adjacency matrix
Mateˊrn(κ22ν+Δ)2νf=WWWsquared exponential(κ22ν−Δ)2ν+4de4κ2Δf=WWW Δ: graph Laplacian WWW: standard Gaussian
Mateˊrnf∼N(0,e−4κ2Δ)f∼N(0,(κ22ν+Δ)−ν)squared exponentialf∼N(0,e−2κ2Δ)
kν(x,x′)=Cνσ2n=1∑∣G∣(κ22ν+λn)−νfn(x)fn(x′) λn,fn: eigenvalues and eigenvectors of graph Laplacian
V. Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Graphs. Artificial Intelligence and Statistics, 2021.
V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Riemannian Manifolds. Advances in Neural Information Processing Systems, 2020.