@inproceedings{
karhadkar2024bounds,
title={Bounds for the smallest eigenvalue of the {NTK} for arbitrary spherical data of arbitrary dimension},
author={Kedar Karhadkar and Michael Murray and Guido Montufar},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=mHVmsy9len}
}
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.
Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension
Kedar Karhadkar Michael Murray Guido Montufar