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

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. SaatΓ§i 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.

TBC.

## Simulating lattice Gaussian fields with the desired covariance structure

## In regression

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 SaatΓ§i (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.

TBC

## Incoming

- Scaling multi-output Gaussian process models with exact inference
- Gaussian Process Latent Variable Model
- Cox Processes (w/ Pyro/GPyTorch Low-Level Interface)
- Latent Function Inference with Pyro + GPyTorch (Low-Level Interface)
- Exact DKL (Deep Kernel Learning) Regression w/ KISS-GP
- Exact GPs with Scalable (GPU) Inference β GPyTorch
- GP Regression with LOVE for Fast Predictive Variances and Sampling

## References

*arXiv:1509.07526 [Math]*, October.

*Journal of the Royal Statistical Society: Series C (Applied Statistics)*46 (1): 171β81.

*Statistics and Computing*9 (4): 265β68.

*Journal of Machine Learning Research*22 (74): 1β6.

*arXiv:1605.09049 [Cs, Stat]*, May.

*Biometrika*91 (1): 211β18.

*Statistical Science*28 (3): 424β46.

*Journal of Statistical Software*55 (9).

*Water Resources Research*29 (8): 2861β69.

*SIAM Journal on Scientific Computing*18 (4): 1088β1107.

*Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics*, 2780β89. PMLR.

*Linear Algebra and Its Applications*173 (August): 19β38.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*37 (2): 424β36.

*arXiv:1710.00751 [Math]*, October.

*arXiv:1710.09254 [Math]*, October.

*Toeplitz and Circulant Matrices: A Review*. Vol. 2.

*Journal of Computational and Graphical Statistics*26 (1): 88β97.

*Linear Algebra and Its Applications*435 (1): 1β59.

*arXiv:1308.0399 [Stat]*, August.

*Monte Carlo Methods and Applications*17 (3).

*arXiv:2003.05554 [Cs, Stat]*, October.

*Mathematical Geosciences*45 (4): 411β35.

*Matrix*2 (2): 1.

*Journal of the Royal Statistical Society. Series B (Statistical Methodology)*63 (2): 325β38.

*Gaussian Markov Random Fields: Theory and Applications*. Monographs on Statistics and Applied Probability 104. Boca Raton: Chapman & Hall/CRC.

*Proceedings of the 27th International Conference on International Conference on Machine Learning*, 927β34. ICMLβ10. Madison, WI, USA: Omnipress.

*Journal of Statistical Software*63 (14).

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*77 (1): 3β33.

*Journal of Computational and Graphical Statistics*26 (1): 108β20.

*Applied Mathematics*7 (17): 2183β94.

*SIAM Journal on Matrix Analysis and Applications*38 (4): 1075β99.

*Biometrika*41 (3/4): 434β49.

*Biometrika*39 (3-4): 309β18.

*Journal of the Royal Statistical Society: Series B (Methodological)*15 (1): 125β39.

*Arkiv FΓΆr Matematik*2 (5): 423β34.

*Scandinavian Actuarial Journal*1952 (1-2): 48β60.

*arXiv:1511.01870 [Cs, Stat]*, November.

*Proceedings of the 32Nd International Conference on International Conference on Machine Learning - Volume 37*, 1775β84. ICMLβ15. Lille, France: JMLR.org.

*Journal of Machine Learning Research*22 (105): 1β47.

*Foundations of Computational Mathematics*16 (3): 577β98.

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