Gaussian processes on lattices

October 30, 2019 — February 28, 2022

dynamical systems
linear algebra
signal processing
state space models
time series
Figure 1

\[ \renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}[1]{\mathbb{I}\{#1\}} \renewcommand{\inner}[2]{\langle #1,#2\rangle} \renewcommand{\Inner}[2]{\left\langle #1,#2\right\rangle} \renewcommand{\norm}[1]{\| #1\|} \renewcommand{\Norm}[1]{\|\langle #1\right\|} \renewcommand{\argmax}{\operatorname{arg max}} \renewcommand{\argmin}{\operatorname{arg min}} \renewcommand{\omp}{\mathop{\mathrm{OMP}}} \]

Gaussian Processes with a stationary kernel are faster if you are working on a grid of points. I have not used this trick, but I understand it involves various simplifications arising from the structure of Gram matrices which end up being Kronecker products of Toeplitz matrices under lexical ordering of the input points, which is long words to describe a fairly simple thing. The keyword to highlight this is Kronecker inference in the ML literature (e.g. Saatçi 2012; Flaxman et al. 2015; A. G. Wilson and Nickisch 2015). Another keyword is circulant embedding, the approximation of a Toeplitz matrix by a circulant matrix, which apparently enables one to leverage fast Fourier transforms to calculate some quantities of interest, and some other nice linear algebra properties besides. That would imply that this method is has its origins in Whittle likelihoods (P. Whittle 1953a; Peter Whittle 1952), if I am not mistaken.

Lattice GP methods complements, perhaps, the trick of filtering Gaussian processes, which can also exploit structure lattice inputs, although the setup is different between these.


1 Simulating lattice Gaussian fields with the desired covariance structure

See GP simulation on lattices.

2 In regression

Figure 2

OK, so what does this allow us to do with posterior inference in GPs? Apparently quite a lot. The KISS-GP method (A. G. Wilson and Nickisch 2015) presumably leverages something similar. So do Saatçi (2012) and Flaxman et al. (2015).

Flaxman et al. (2015) deals with the lattice structure trick in a non-Gaussian likelihood setting using Laplace approximation.


3 Incoming

4 References

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