Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension

Abstract

Kernel methods such as kernel ridge regression and Gaussian process regressions with Matern type kernels have been increasingly used, in particular, to fit potential energy surfaces (PES) and density functionals, and for materials informatics. When the dimensionality of the feature space is high, these methods are used with necessarily sparse data. In this regime, the optimal length parameter of a Matern-type kernel tends to become so large that the method effectively degenerates into a low-order polynomial regression and therefore loses any advantage over such regression. This is demonstrated theoretically as well as numerically on the examples of six- and fifteen-dimensional molecular PES using squared exponential and simple exponential kernels. The results shed additional light on the success of polynomial approximations such as PIP for medium size molecules and on the importance of orders-of-coupling based models for preserving the advantages of kernel methods with Matern type kernels or on the use of physically-motivated (reproducing) kernels.

References in corpus (1)

• Data-driven kinetic energy density fitting for orbital-free DFT: linear vs Gaussian process regression

Cited by in corpus (3)

• A machine-learned kinetic energy model for light weight metals and compounds of group III-V elements
• Machine learning the screening factor in the soft bond valence approach for rapid crystal structure estimation
• Machine learning-guided construction of an analytic kinetic energy functional for orbital free density functional theory