Abstract
We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an \(N\times N\) Wigner matrix. Suppose that k randomly chosen entries of the matrix are resampled, resulting in another realization of the Wigner matrix with top eigenvector \(v^{[k]}\). We prove that, with high probability, when \(k \ll N^{5/3-o(1)}\), then v and \(v^{[k]}\) are almost collinear and when \(k\gg N^{5/3}\), then \(v^{[k]}\) is almost orthogonal to v.
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Acknowledgements
We would like to thank Jaehun Lee for pointing out a mistake in the proof of Lemma 3 in an early version of this paper. We also would like to thank the referees for their valuable reports. Funding was provided by Ministerio de Economía y Competitividad (Grant No. MTM2015-67304-P).
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Gábor Lugosi was supported by the Spanish Ministry of Economy and Competitiveness, Grant PGC2018-101643-B-I00; “High-dimensional problems in structured probabilistic models—Ayudas Fundación BBVA a Equipos de Investigación Cientifica 2017”; and Google Focused Award “Algorithms and Learning for AI”. Charles Bordenave was supported by by the research grants ANR-14-CE25-0014 and ANR-16-CE40-0024-01. Nikita Zhivotovskiy was supported by RSF Grant No. 18-11-00132.
This work was prepared while Nikita Zhivotovskiy was a postdoctoral fellow at the department of Mathematics, Technion I.I.T. and researcher at National University Higher School of Economics. Now at Google Research, Brain Team.
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Bordenave, C., Lugosi, G. & Zhivotovskiy, N. Noise sensitivity of the top eigenvector of a Wigner matrix. Probab. Theory Relat. Fields 177, 1103–1135 (2020). https://doi.org/10.1007/s00440-020-00970-1
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DOI: https://doi.org/10.1007/s00440-020-00970-1