## The Annals of Statistics

### Phase transition in the spiked random tensor with Rademacher prior

Wei-Kuo Chen

#### 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.

#### Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2734-2756.

Dates
Revised: August 2018
First available in Project Euclid: 3 August 2019

https://projecteuclid.org/euclid.aos/1564797862

Digital Object Identifier
doi:10.1214/18-AOS1763

Mathematical Reviews number (MathSciNet)
MR3988771

Zentralblatt MATH identifier
07114927

#### Citation

Chen, Wei-Kuo. Phase transition in the spiked random tensor with Rademacher prior. Ann. Statist. 47 (2019), no. 5, 2734--2756. doi:10.1214/18-AOS1763. https://projecteuclid.org/euclid.aos/1564797862

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#### Supplemental materials

• Supplement to “Phase transition in the spiked random tensor with Rademacher prior”. The proofs of Theorems 4.1, 4.2, 4.3 and Proposition 4.1 are provided in detail in the Supplementary Material [18]. In addition, the convergence of the free energies $AF_{N}$ and $L_{N}$ defined respectively by (3.1) and (3.10) are established.