Reverse Engineering the Neural Tangent Kernel

TL;DR

This paper demonstrates how to convert high-performing kernels into neural network architectures with similar properties, enabling principled design of neural networks based on kernel functions.

Contribution

It proves that any positive-semidefinite dot-product kernel can be realized as the NNGP or neural tangent kernel of a single-hidden-layer network with an appropriate activation function.

Findings

Constructive proof of kernel-to-network realization.

Numerical verification of the theoretical construction.

Demonstrated utility as a neural network design tool.

Abstract

The development of methods to guide the design of neural networks is an important open challenge for deep learning theory. As a paradigm for principled neural architecture design, we propose the translation of high-performing kernels, which are better-understood and amenable to first-principles design, into equivalent network architectures, which have superior efficiency, flexibility, and feature learning. To this end, we constructively prove that, with just an appropriate choice of activation function, any positive-semidefinite dot-product kernel can be realized as either the NNGP or neural tangent kernel of a fully-connected neural network with only one hidden layer. We verify our construction numerically and demonstrate its utility as a design tool for finite fully-connected networks in several experiments.