Statistical Query Lower Bounds for Tensor PCA

In the Tensor PCA problem introduced by Richard and Montanari (2014), one is given a dataset consisting of n samples T1:n of i.i.d. Gaussian tensors of order k with the promise that mathbbET1 is a rank-1 tensor and |mathbbE T1| = 1. The goal is to estimate mathbbE T1. This problem exhibits a large conjectured hard phase when k>2: When d lesssim n ll d^frack2 it is information theoretically possible to estimate mathbbE T1, but no polynomial time estimator is known. We provide a sharp analysis of the optimal sample complexity in the Statistical Query (SQ) model and show that SQ algorithms with polynomial query complexity not only fail to solve Tensor PCA in the conjectured hard phase, but also have a strictly sub-optimal sample complexity compared to some polynomial time estimators such as the Richard-Montanari spectral estimator. Our analysis reveals that the optimal sample complexity in the SQ model depends on whether mathbbE T1 is symmetric or not. For symmetric, even order tensors, we also isolate a sample size regime in which it is possible to test if mathbbE T1 = 0 or mathbbET1 neq 0 with polynomially many queries but not estimate mathbbET1. Our proofs rely on the Fourier analytic approach of Feldman, Perkins and Vempala (2018) to prove sharp SQ lower bounds.