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 (8)

• Active learning of reactive Bayesian force fields: Application to heterogeneous hydrogen-platinum catalysis dynamics
• Permutationally invariant polynomial regression for energies and gradients, using reverse differentiation, achieves orders of magnitude speed-up with high precision compared to other machine learning methods
• Data-driven kinetic energy density fitting for orbital-free DFT: linear vs Gaussian process regression
• Random Sampling High Dimensional Model Representation Gaussian Process Regression (RS-HDMR-GPR) for representing multidimensional functions with machine-learned lower-dimensional terms allowing insight with a general method
• Neural network with optimal neuron activation functions based on additive Gaussian process regression
• Easy representation of multivariate functions with low-dimensional terms via Gaussian process regression kernel design: applications to machine learning of potential energy surfaces and kinetic energy densities from sparse data
• The loss of the property of locality of the kernel in high-dimensional Gaussian process regression on the example of the fitting of molecular potential energy surfaces
• Orders-of-coupling representation with a single neural network with optimal neuron activation functions and without nonlinear parameter optimization

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