Eigen Analysis of Conjugate Kernel and Neural Tangent Kernel

Xiangchao Li, Xiao Han, Qing Yang
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:34490-34508, 2025.

Abstract

In this paper, we investigate deep feedforward neural networks with random weights. The input data matrix $\boldsymbol{X}$ is drawn from a Gaussian mixture model. We demonstrate that certain eigenvalues of the conjugate kernel and neural tangent kernel may lie outside the support of their limiting spectral measures in the high-dimensional regime. The existence and asymptotic positions of such isolated eigenvalues are rigorously analyzed. Furthermore, we provide a precise characterization of the entrywise limit of the projection matrix onto the eigenspace associated with these isolated eigenvalues. Our findings reveal that the eigenspace captures inherent group features present in $\boldsymbol{X}$. This study offers a quantitative analysis of how group features from the input data evolve through hidden layers in randomly weighted neural networks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-li25u, title = {Eigen Analysis of Conjugate Kernel and Neural Tangent Kernel}, author = {Li, Xiangchao and Han, Xiao and Yang, Qing}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {34490--34508}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/li25u/li25u.pdf}, url = {https://proceedings.mlr.press/v267/li25u.html}, abstract = {In this paper, we investigate deep feedforward neural networks with random weights. The input data matrix $\boldsymbol{X}$ is drawn from a Gaussian mixture model. We demonstrate that certain eigenvalues of the conjugate kernel and neural tangent kernel may lie outside the support of their limiting spectral measures in the high-dimensional regime. The existence and asymptotic positions of such isolated eigenvalues are rigorously analyzed. Furthermore, we provide a precise characterization of the entrywise limit of the projection matrix onto the eigenspace associated with these isolated eigenvalues. Our findings reveal that the eigenspace captures inherent group features present in $\boldsymbol{X}$. This study offers a quantitative analysis of how group features from the input data evolve through hidden layers in randomly weighted neural networks.} }
Endnote
%0 Conference Paper %T Eigen Analysis of Conjugate Kernel and Neural Tangent Kernel %A Xiangchao Li %A Xiao Han %A Qing Yang %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-li25u %I PMLR %P 34490--34508 %U https://proceedings.mlr.press/v267/li25u.html %V 267 %X In this paper, we investigate deep feedforward neural networks with random weights. The input data matrix $\boldsymbol{X}$ is drawn from a Gaussian mixture model. We demonstrate that certain eigenvalues of the conjugate kernel and neural tangent kernel may lie outside the support of their limiting spectral measures in the high-dimensional regime. The existence and asymptotic positions of such isolated eigenvalues are rigorously analyzed. Furthermore, we provide a precise characterization of the entrywise limit of the projection matrix onto the eigenspace associated with these isolated eigenvalues. Our findings reveal that the eigenspace captures inherent group features present in $\boldsymbol{X}$. This study offers a quantitative analysis of how group features from the input data evolve through hidden layers in randomly weighted neural networks.
APA
Li, X., Han, X. & Yang, Q.. (2025). Eigen Analysis of Conjugate Kernel and Neural Tangent Kernel. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:34490-34508 Available from https://proceedings.mlr.press/v267/li25u.html.

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