# Invariances of random fields paths, with applications in Gaussian Process Regression

arXiv: https://arxiv.org/abs/1308.1359 · HTML: https://mlsanity.com/paper/1308.1359/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2013.

## Built on by (most cited first, 2 of 2)

- [Data augmentation in Bayesian neural networks and the cold posterior effect](https://mlsanity.com/paper/2106.05586/) (2021)
- [Omnipresent Yet Overlooked: Heat Kernels in Combinatorial Bayesian Optimization](https://mlsanity.com/paper/2510.26633/) (2025)

## Abstract (arXiv)

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Lo`eve isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process regression.