[Paper Review] Invariances of random fields paths, with applications in Gaussian Process Regression

David Ginsbourger, Olivier Roustant+1 more|arXiv (Cornell University)|Aug 6, 2013

Gaussian Processes and Bayesian Inference41 references3 citations

TL;DR

This paper establishes a theoretical framework linking pathwise invariances in random fields—such as additivity, symmetry, or sparsity—to invariance properties of their covariance kernels. It shows that for centred square-integrable random fields, path invariance under linear combinations of composition operators is equivalent to argumentwise invariance of the kernel, with extensions to Gaussian processes via the Loève isometry, enabling improved Gaussian process regression through structured kernel design.

ABSTRACT

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.

Motivation & Objective

• To characterize pathwise invariances in random fields using covariance kernel structure.
• To extend existing results on additive and group-invariant kernels to a broader class of operators.
• To demonstrate how structural priors (e.g., sparsity, symmetry) can be encoded in kernels to improve Gaussian process regression.
• To provide a theoretical foundation for kernel design that enforces desired functional path properties.
• To show that invariance assumptions enhance model accuracy even when the true function is not perfectly invariant.

Proposed method

• Proposes a general framework linking path invariance under linear combinations of composition operators to argumentwise invariance of the covariance kernel.
• Uses the Loève isometry to extend results from second-order random fields to Gaussian processes, enabling invariance under differential and integral operators.
• Characterizes additive paths in high-dimensional random fields via kernel decomposition, linking to ANOVA-type and sparse kernels.
• Applies the theory to Gaussian process regression by designing kernels that enforce structural priors such as symmetry, additivity, and sparsity.
• Employs maximum likelihood estimation for hyperparameters and evaluates models using log-likelihood, RMSE, and Q² metrics.
• Demonstrates that kernel-based invariance leads to better predictive performance than standard kernels, even when the true function is only approximately invariant.

Experimental results

Research questions

• RQ1 How can pathwise invariances such as additivity or symmetry in random fields be characterized through their covariance kernel?
• RQ2 What conditions on the covariance kernel ensure that sample paths are invariant under a given linear operator?
• RQ3 To what extent can invariance in the kernel structure improve predictive performance in Gaussian process regression?
• RQ4 How does the Loève isometry enable the extension of path invariance results from second-order to Gaussian processes?
• RQ5 Can structured kernels that enforce sparsity or symmetry outperform standard kernels in high-dimensional function approximation?

Key findings

• A centred random field has paths invariant under a linear combination of composition operators if and only if the kernel is invariant under the same transformation applied to its arguments.
• The proposed framework generalizes prior results on additive and group-invariant kernels, providing a complete characterization of kernels leading to additive paths in high-dimensional settings.
• In Gaussian processes, invariance under broader operators (e.g., differential or integral operators) can be achieved via the Loève isometry, linking path invariance to kernel structure.
• Models using sparse kernels derived from sensitivity analysis (e.g., $k_{spa}$) achieved the best predictive performance, with Q² = 0.71 and RMSE = 0.74, outperforming standard $k_{gauss}$ and $k_{add}$.
• Even when the true function is not perfectly invariant, assuming structural priors such as sparsity significantly improves model accuracy by mitigating the curse of dimensionality.
• The log-likelihood improved from -45.73 (for $k_{gauss}$) to -1.48 (for $k_{anova}$), indicating a substantial gain in model fit through kernel design.