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arXiv · 2405.14630

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

Kedar Karhadkar·Michael Murray·Guido Montúfar
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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.

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Kedar Karhadkar, Michael Murray, Guido Montúfar. 2024-05-23. Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension. https://arxiv.org/abs/2405.14630

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