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.

翻译：高斯过程（Gaussian Processes, GPs）是一种通用方法，支持多种面向动力学与控制的机器学习范式。在高斯过程中，高斯性假设体现在两个维度：底层再生核希尔伯特空间的正定核被用于构造函数空间上的高斯分布协方差，而测量噪声（即数据污染）通常被建模为独立同分布加性高斯噪声。本文中，我们放宽了后者高斯性假设，即考虑带有独立同分布非高斯加性测量噪声的核岭回归。为应用常规核技巧，我们借助多项式混沌展开（由Norbert Wiener提出的有限方差随机变量级数展开）表征不确定性。我们推导并讨论了韦纳核回归问题的解析$\mathcal{L}^2$解。以多项式系统作为数值示例，我们证明了该方法能够解耦认知不确定性与偶然不确定性的影响。