Phase transition in the spiked random tensor with Rademacher prior
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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. (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 (2017) proved that there exists a β_p'<β_p such that any statistical hypothesis test can not 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.
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