Correspondence of NNGP Kernel and the Matern Kernel

TL;DR

This paper investigates the NNGP kernel's properties, its relation to the Matern kernel, and compares their practical performance, concluding that the Matern kernel is generally more flexible and preferable for practical Gaussian process regression tasks.

Contribution

It reveals the close correspondence between the NNGP and Matern kernels and evaluates their practical performance, highlighting the Matern kernel's advantages.

Findings

NNGP kernels require normalization for validity.

Predictions from NNGP are inflexible across hyperparameters.

Matern kernel outperforms NNGP in practical benchmarks.

Abstract

Kernels representing limiting cases of neural network architectures have recently gained popularity. However, the application and performance of these new kernels compared to existing options, such as the Matern kernel, is not well studied. We take a practical approach to explore the neural network Gaussian process (NNGP) kernel and its application to data in Gaussian process regression. We first demonstrate the necessity of normalization to produce valid NNGP kernels and explore related numerical challenges. We further demonstrate that the predictions from this model are quite inflexible, and therefore do not vary much over the valid hyperparameter sets. We then demonstrate a surprising result that the predictions given from the NNGP kernel correspond closely to those given by the Matern kernel under specific circumstances, which suggests a deep similarity between overparameterized deep neural networks and the Matern kernel. Finally, we demonstrate the performance of the NNGP kernel as compared to the Matern kernel on three benchmark data cases, and we conclude that for its flexibility and practical performance, the Matern kernel is preferred to the novel NNGP in practical applications.