On automatic differentiation for the Mat\'ern covariance

TL;DR

This paper develops accurate and efficient methods for computing derivatives of the Matérn covariance function with respect to its smoothness parameter, addressing a key challenge in Gaussian process modeling.

Contribution

It introduces new series expansions for the modified Bessel function of the second kind, enabling robust derivatives with respect to the order in Gaussian processes.

Findings

Series expansions provide high accuracy for derivatives.

Complex step method outperforms finite differences in efficiency.

Enhanced derivatives facilitate better optimization in Gaussian processes.

Abstract

To target challenges in differentiable optimization we analyze and propose strategies for derivatives of the Mat\'ern kernel with respect to the smoothness parameter. This problem is of high interest in Gaussian processes modelling due to the lack of robust derivatives of the modified Bessel function of second kind with respect to order. In the current work we focus on newly identified series expansions for the modified Bessel function of second kind valid for complex orders. Using these expansions we obtain highly accurate results using the complex step method. Furthermore, we show that the evaluations using the recommended expansions are also more efficient than finite differences.