Open Access
April 2020 Fundamental limits of detection in the spiked Wigner model
Ahmed El Alaoui, Florent Krzakala, Michael Jordan
Ann. Statist. 48(2): 863-885 (April 2020). DOI: 10.1214/19-AOS1826


We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the nonspiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a “spectral” nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. Further information on the performance of the optimal test is also provided. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington–Kirkpatrick spin-glass model.


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Ahmed El Alaoui. Florent Krzakala. Michael Jordan. "Fundamental limits of detection in the spiked Wigner model." Ann. Statist. 48 (2) 863 - 885, April 2020.


Received: 1 October 2017; Revised: 1 February 2019; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241572
MathSciNet: MR4102679
Digital Object Identifier: 10.1214/19-AOS1826

Primary: 62H25
Secondary: 60F05 , 60G15 , 62H15

Keywords: contiguity , Hypothesis testing , random matrix models , replica–symmetry , Sherrington–Kirkpatrick model , spin–glasses

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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