## Abstract

We consider the problem of detecting a deformation from a symmetric Gaussian random p-tensor (p ≥ 3) 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 β_{p} so that when the signal-to-noise ratio exceeds β_{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 β'_{p} < β_{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 β'_{p}. In this work, we show that β_{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 β_{p} as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure p-spin mean-field spin glass model.

Original language | English (US) |
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Pages (from-to) | 2734-2736 |

Number of pages | 3 |

Journal | Annals of Statistics |

Volume | 47 |

Issue number | 5 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Funding Information:Received December 2017; revised August 2018. 1Supported in part by NSF DMS-1642207, NSF DMS-1752184 and Hong Kong Research Grants Council GRF-14302515. MSC2010 subject classifications. Primary 93E10; secondary 60K35, 82B44. Key words and phrases. BBP transition, signal detection, Parisi formula, replica symmetry breaking, spin glass, spiked tensor.

Publisher Copyright:

© Institute of Mathematical Statistics, 2019.

## Keywords

- BBP transition
- Parisi formula
- Replica symmetry breaking
- Signal detection
- Spiked tensor
- Spin glass