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

David Ginsbourger (1) , Olivier Roustant (2, 3) , Nicolas Durrande (4, 5)

1 IMSV - Institute of Mathematical Statistics and Actuarial Science [Bern]
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 DEMO-ENSMSE - Département Décision en Entreprise : Modélisation, Optimisation
4 Department of Computer Science
5 Sheffield Institute for Translational Neuroscience

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.

Mots clés: Structural priors; Covariance kernels; Composition operators; RKHS; Bayesian function learning; Structural priors.

Domaines: Modélisation et simulation; Statistiques [math.ST]; Théorie [stat.TH]; Machine Learning [stat.ML]; Méthodologie [stat.ME]; Probabilités [math.PR]

David Ginsbourger, Olivier Roustant, Nicolas Durrande. Invariances of random fields paths, with applications in Gaussian Process Regression. 2013. ⟨hal-00850436⟩