Abstract

We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor. We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the planted vector: (i) a uniformly sampled unit vector, (ii) i.i.d. $\pm 1$ entries, and (iii) a sparse vector where a constant fraction $\rho$ of entries are i.i.d. $\pm 1$ and the rest are zero. For each of these cases, our upper and lower bounds match up to a $1+o(1)$ factor as the order $d$ of the tensor becomes large. For sparse signals (iii), our bounds are also asymptotically tight in the sparse limit $\rho\to 0$ for any fixed $d$ (including the $d=2$ case of sparse PCA). Our upper bounds for (i) demonstrate a phenomenon reminiscent of the work of Baik, Ben Arous and Péché: an ‘eigenvalue’ of a perturbed tensor emerges from the bulk at a strictly lower signal-to-noise ratio than when the perturbation itself exceeds the bulk; we quantify the size of this effect. We also provide some general results for larger classes of priors. In particular, the large $d$ asymptotics of the threshold location differs between problems with discrete priors versus c