  We consider the problem of detecting a deformation from a symmetric Gaussian
random $p$-tensor $(p\geq 3)$ 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 $\beta_p$ so that when the signal-to-noise ratio exceeds
$\beta_p$, 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 $\beta_p'<\beta_p$
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 $\beta_p'$. In this work,
we show that $\beta_p$ 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 $\beta_p$ as the critical temperature,
distinguishing the high and low temperature behavior, of the Ising pure
$p$-spin mean-field spin glass model.
