Phase transition in the spiked random tensor with rademacher prior

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.