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October 2019 Phase transition in the spiked random tensor with Rademacher prior
Wei-Kuo Chen
Ann. Statist. 47(5): 2734-2756 (October 2019). DOI: 10.1214/18-AOS1763


We consider the problem of detecting a deformation from a symmetric Gaussian random $p$-tensor $(p\geq3)$ with a rank-one spike sampled from the Rademacher prior. Recently, in Lesieur et al. (Barbier, Krzakala, Macris, Miolane and Zdeborová (2017)), it was proved that there exists a critical threshold $\beta_{p}$ so that when the signal-to-noise ratio exceeds $\beta_{p}$, one can distinguish the spiked and unspiked tensors and weakly recover the prior via the minimal mean-square-error method. On the other side, Perry, Wein and Bandeira (Perry, Wein and Bandeira (2017)) proved that there exists a $\beta_{p}'<\beta_{p}$ such that any statistical hypothesis test cannot distinguish these two tensors, in the sense that their total variation distance asymptotically vanishes, when the signa-to-noise ratio is less than $\beta_{p}'$. In this work, we show that $\beta_{p}$ is indeed the critical threshold that strictly separates the distinguishability and indistinguishability between the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure $p$-spin model with Ising spin, arising initially from the field of spin glasses. In particular, we identify the signal-to-noise criticality $\beta_{p}$ as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure $p$-spin mean-field spin glass model.


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Wei-Kuo Chen. "Phase transition in the spiked random tensor with Rademacher prior." Ann. Statist. 47 (5) 2734 - 2756, October 2019.


Received: 1 December 2017; Revised: 1 August 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114927
MathSciNet: MR3988771
Digital Object Identifier: 10.1214/18-AOS1763

Primary: 93E10
Secondary: 60K35 , 82B44

Keywords: BBP transition , Parisi formula , replica symmetry breaking , signal detection , spiked tensor , Spin glass

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 5 • October 2019
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