Characterizing the spectrum of the NTK via a power series expansionDownload PDF

Published: 01 Feb 2023, 19:21, Last Modified: 28 Feb 2023, 02:03ICLR 2023 posterReaders: Everyone
Keywords: neural tangent kernel, power series, Hermite coefficient, activation function, spectrum, input Gram matrix
TL;DR: We characterize the NTK spectrum via a power series representation in terms of the Hermite coefficients of the activation function, the depth, and the effective rank of the input Gram.
Abstract: Under mild conditions on the network initialization we derive a power series expansion for the Neural Tangent Kernel (NTK) of arbitrarily deep feedforward networks in the infinite width limit. We provide expressions for the coefficients of this power series which depend on both the Hermite coefficients of the activation function as well as the depth of the network. We observe faster decay of the Hermite coefficients leads to faster decay in the NTK coefficients and explore the role of depth. Using this series, first we relate the effective rank of the NTK to the effective rank of the input-data Gram. Second, for data drawn uniformly on the sphere we study the eigenvalues of the NTK, analyzing the impact of the choice of activation function. Finally, for generic data and activation functions with sufficiently fast Hermite coefficient decay, we derive an asymptotic upper bound on the spectrum of the NTK.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics
Submission Guidelines: Yes
Please Choose The Closest Area That Your Submission Falls Into: Theory (eg, control theory, learning theory, algorithmic game theory)
17 Replies