Phase transition in the spiked random tensor with Rademacher prior

We consider the symmetric Gaussian random $p$-tensor deformed by a rank-one spike sampled from the Rademacher prior. We show that there exists a critical threshold so that if the signal-to-noise ratio is greater than this criticality, it is possible to distinguish the spiked and unspiked tensors and to recover the prior in a weak sense. In contrast, if the signal-to-noise ratio is less than the criticality, it is proved that the two tensors are indistinguishable and weak recovery is impossible. 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, it is identified that the signal-to-noise criticality is the critical temperature, distinguishing the high and low temperature behavior, of the pure $p$-spin model.