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

Part of Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track

Authors

Kedar Karhadkar, Michael Murray, Guido Montúfar

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.