Distribution of Gaussian Process Arc Lengths

Justin Bewsher, Alessandra Tosi, Michael Osborne, Stephen Roberts
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:1412-1420, 2017.

Abstract

We present the first treatment of the arc length of the GP with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical consideration of higher dimensional problems. We fill the gap in the existing literature by deriving the moments of the arc length for a stationary GP with multiple output dimensions. A new method is used to derive the mean of a one-dimensional GP over a finite interval, by considering the distribution of the arc length integrand. This technique is used to derive an approximate distribution over the arc length of a vector valued GP in $\mathbbR^n$ by moment matching the distribution. Numerical simulations confirm our theoretical derivations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-bewsher17a, title = {{Distribution of Gaussian Process Arc Lengths}}, author = {Bewsher, Justin and Tosi, Alessandra and Osborne, Michael and Roberts, Stephen}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {1412--1420}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/bewsher17a/bewsher17a.pdf}, url = {https://proceedings.mlr.press/v54/bewsher17a.html}, abstract = { We present the first treatment of the arc length of the GP with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical consideration of higher dimensional problems. We fill the gap in the existing literature by deriving the moments of the arc length for a stationary GP with multiple output dimensions. A new method is used to derive the mean of a one-dimensional GP over a finite interval, by considering the distribution of the arc length integrand. This technique is used to derive an approximate distribution over the arc length of a vector valued GP in $\mathbbR^n$ by moment matching the distribution. Numerical simulations confirm our theoretical derivations.} }
Endnote
%0 Conference Paper %T Distribution of Gaussian Process Arc Lengths %A Justin Bewsher %A Alessandra Tosi %A Michael Osborne %A Stephen Roberts %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-bewsher17a %I PMLR %P 1412--1420 %U https://proceedings.mlr.press/v54/bewsher17a.html %V 54 %X We present the first treatment of the arc length of the GP with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical consideration of higher dimensional problems. We fill the gap in the existing literature by deriving the moments of the arc length for a stationary GP with multiple output dimensions. A new method is used to derive the mean of a one-dimensional GP over a finite interval, by considering the distribution of the arc length integrand. This technique is used to derive an approximate distribution over the arc length of a vector valued GP in $\mathbbR^n$ by moment matching the distribution. Numerical simulations confirm our theoretical derivations.
APA
Bewsher, J., Tosi, A., Osborne, M. & Roberts, S.. (2017). Distribution of Gaussian Process Arc Lengths. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:1412-1420 Available from https://proceedings.mlr.press/v54/bewsher17a.html.

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