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

arXiv: https://arxiv.org/abs/2311.10790 · HTML: https://mlsanity.com/paper/2311.10790/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 45 references that are in the map.
- First posted on arXiv 17 Nov 2023.

## Builds on (most cited first, 8 of 8)

- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Understanding Machine-learned Density Functionals](https://mlsanity.com/paper/1404.1333/) (2014)
- [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](https://mlsanity.com/paper/2012.02704/) (2020)
- [Neural network with optimal neuron activation functions based on additive Gaussian process regression](https://mlsanity.com/paper/2301.05567/) (2023)
- [Understanding Kernel Ridge Regression: Common behaviors from simple functions to density functionals](https://mlsanity.com/paper/1501.03854/) (2015)
- [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](https://mlsanity.com/paper/2211.11170/) (2022)
- [Orders-of-coupling representation with a single neural network with optimal neuron activation functions and without nonlinear parameter optimization](https://mlsanity.com/paper/2302.12013/) (2023)

## Built on by (most cited first, 3 of 3)

- [Machine learning-guided construction of an analytic kinetic energy functional for orbital free density functional theory](https://mlsanity.com/paper/2502.05411/) (2025)
- [Gaussian Process Regression -- Neural Network Hybrid with Optimized Redundant Coordinates](https://mlsanity.com/paper/2509.08457/) (2025)
- [Distributed Quantum Gaussian Processes for Multi-Agent Systems](https://mlsanity.com/paper/2602.15006/) (2026)

## Abstract (arXiv)

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.