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

Kedar Karhadkar, Michael Murray, Guido Montufar

Keywords: neural tangent kernel, initialization, minimum eigenvalue, smallest eigenvalue, low-dimensional, hemisphere transform, spherical harmonics, separated

TL;DR: We bound the smallest eigenvalue of the NTK without distributional assumptions on the data.

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 distance between data points: 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.

Primary Area: Learning theory

Submission Number: 12761