PhD viva

Gaussian Processes

 and

and Statistical Decision-making

in Non-Euclidean spaces

Alexander Terenin

https://avt.im/ · @avt_im

Gaussian Processes

Statistical Decision-making

Pathwise Conditioning

prior term + data term = posterior Gaussian process

$$ \htmlData{class=fragment,fragment-index=2} { (f\given y)(\.) = } \htmlData{class=fragment,fragment-index=0} { \mathrlap{f(\.)} } \htmlData{class=fragment,fragment-index=3} { \ubr{\phantom{f(\.)}}{\c{O}(N_*^3)} } \htmlData{class=fragment,fragment-index=1} { + \m{K}_{(\.)\v{x}} } \htmlData{class=fragment,fragment-index=1} { \mathrlap{\m{K}_{\v{x}\v{x}}^{-1}} } \htmlData{class=fragment,fragment-index=3} { \ubr{\phantom{\m{K}_{\v{x}\v{x}}^{-1}}}{\c{O}(N^3)} } \htmlData{class=fragment,fragment-index=1} { (\v{y} - f(\v{x})) } $$

Riemannian Matérn Gaussian Processes

Riemannian Matérn Gaussian Processes

(a) Ground truth

(b) Posterior mean

(c) Standard deviation

(d) Posterior sample

Graph Matérn Gaussian Processes

(a) Predictive mean

(b) Standard deviation

Graph Matérn Gaussian Processes

(a) Predictive mean

(b) Standard deviation

Graph Matérn Gaussian Processes

Gaussian Vector Fields

Thank you!

https://avt.im/· @avt_im

J. T. Wilson,* V. Borovitskiy,* P. Mostowsky,* A. Terenin,* M. P. Deisenroth. Efficiently Sampling Functions from Gaussian Process Posteriors. International Conference on Machine Learning, 2020.

J. T. Wilson,* V. Borovitskiy,* P. Mostowsky,* A. Terenin,* M. P. Deisenroth. Pathwise Conditioning of Gaussian Process. Journal of Machine Learning Research, 2021.

V. Borovitskiy,* P. Mostowsky,* A. Terenin,* M. P. Deisenroth. Matérn Gaussian Processes on Riemannian Manifolds. Advances in Neural Information Processing Systems, 2020.

V. Borovitskiy,* I. Azangulov,* P. Mostowsky,* A. Terenin,* M. P. Deisenroth, N. Durrande. Matérn Gaussian Processes on Graphs. Artificial Intelligence and Statistics, 2021.

M. J. Hutchinson,* A. Terenin,* V. Borovitskiy,* S. Takao,* Y. W. Teh, M. P. Deisenroth. Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels. Advances in Neural Information Processing Systems, 2021.

N. Jaquier,* V. Borovitskiy,* A. Smolensky, A. Terenin, T. Asfour, L. Rozo. Geometry-aware Bayesian Optimization in Robotics using Riemannian Matérn Kernels. Conference on Robot Learning, 2021.

*Equal contribution