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 βp so that when the signal-to-noise ratio exceeds βp, 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 < βp 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 βp 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 βp as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure p-spin mean-field spin glass model.
Bibliographical noteFunding 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.
© Institute of Mathematical Statistics, 2019.
- BBP transition
- Parisi formula
- Replica symmetry breaking
- Signal detection
- Spiked tensor
- Spin glass