Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices

Aro Lee, Ji Oon Lee
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:33322-33351, 2025.

Abstract

We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik-Ben Arous-Péché (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy-Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-lee25t, title = {Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices}, author = {Lee, Aro and Lee, Ji Oon}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {33322--33351}, 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/lee25t/lee25t.pdf}, url = {https://proceedings.mlr.press/v267/lee25t.html}, abstract = {We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik-Ben Arous-Péché (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy-Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes.} }
Endnote
%0 Conference Paper %T Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices %A Aro Lee %A Ji Oon Lee %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-lee25t %I PMLR %P 33322--33351 %U https://proceedings.mlr.press/v267/lee25t.html %V 267 %X We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik-Ben Arous-Péché (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy-Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes.
APA
Lee, A. & Lee, J.O.. (2025). Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:33322-33351 Available from https://proceedings.mlr.press/v267/lee25t.html.

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