Invariances of random fields paths, with applications in Gaussian Process Regression
This work provides theoretical insights into Gaussian process regression, but it appears incremental as it builds on existing covariance-based methods without introducing a new paradigm.
The authors tackled the problem of understanding pathwise invariances in random fields by linking them to covariance kernel properties, and they extended these results to Gaussian processes using the Loève isometry, illustrating applications like symmetric or sparse paths.
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 Loève 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.