Wiener Chaos in Kernel Regression: Towards Untangling Aleatoric and Epistemic Uncertainty

Gaussian Processes (GPs) are a versatile method that enables different approaches towards learning for dynamics and control. Gaussianity assumptions appear in two dimensions in GPs: The positive semi-definite kernel of the underlying reproducing kernel Hilbert space is used to construct the co-variance of a Gaussian distribution over functions, while measurement noise (i.e. data corruption) is usually modeled as i.i.d. additive Gaussian. In this note, we relax the latter Gaussianity assumption, i.e., we consider kernel ridge regression with additive i.i.d. non-Gaussian measurement noise. To apply the usual kernel trick, we rely on the representation of the uncertainty via polynomial chaos expansions, which are series expansions for random variables of finite variance introduced by Norbert Wiener. We derive and discuss the analytic $\mathcal{L}^2$ solution to the arising Wiener kernel regression. Considering a polynomial system as numerical example, we show that our approach allows to untangle the effects of epistemic and aleatoric uncertainties.