Statistical limits of spiked tensor models

TL;DR

This paper investigates the fundamental statistical limits for detecting and estimating rank-one signals in symmetric Gaussian tensors, providing tight bounds and new methodological insights for various prior distributions.

Contribution

It establishes tight upper and lower bounds on the signal-to-noise ratio for different priors, introduces a novel second moment method improvement, and compares discrete versus continuous prior asymptotics.

Findings

Bounds match up to a 1+o(1) factor as tensor order increases.

For sparse signals, bounds are tight in the sparsity limit as 0.

Replica predictions conjecturally give exact thresholds for fixed tensor order.

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 P\'ech\'e: 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 continuous priors. Finally, for priors (i) and (ii) we carry out the replica prediction from statistical physics, which is conjectured to give the exact information-theoretic threshold for any fixed $d$. Of independent interest, we introduce a new improvement to the second moment method for contiguity, on which our lower bounds are based. Our technique conditions away from rare `bad' events that depend on interactions between the signal and noise. This enables us to close $\sqrt{2}$-factor gaps present in several previous works.