Phase transition in the spiked random tensor with Rademacher prior
The Annals of Statistics · 2019 · Volume 47 · Issue 5
Abstract
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.
Keywords
60K35, 82B44, 93E10, BBP transition, Parisi formula, replica symmetry breaking, signal detection, spiked tensor, Spin glass
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