Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension

TL;DR

This paper derives bounds on the smallest eigenvalue of the neural tangent kernel for arbitrary spherical data of any dimension, removing previous high-dimensional and distributional assumptions, using a novel hemisphere transform approach.

Contribution

It introduces bounds for the NTK's smallest eigenvalue that apply to fixed-dimensional data without distributional assumptions, expanding theoretical understanding.

Findings

Bounds hold with high probability for fixed data dimension

Results do not require distributional assumptions on data

Applicable to arbitrary spherical data of any dimension

Abstract

Bounds on the smallest eigenvalue of the neural tangent kernel (NTK) are a key ingredient in the analysis of neural network optimization and memorization. However, existing results require distributional assumptions on the data and are limited to a high-dimensional setting, where the input dimension $d_0$ scales at least logarithmically in the number of samples $n$. In this work we remove both of these requirements and instead provide bounds in terms of a measure of the collinearity of the data: notably these bounds hold with high probability even when $d_0$ is held constant versus $n$. We prove our results through a novel application of the hemisphere transform.