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 .p so that when the signal-to-noise ratio exceeds .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 ..p<.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 ..p. In this work, we show that .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 .p as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure p-spin mean-field spin glass model.