Tensor Prediction, Rademacher Complexity and Random 3-XOR
Here we study the tensor prediction problem, where the goal is to accurately predict the entries of a low rank, third-order tensor (with noise) given as few observations as possible. We give algorithms based on the sixth level of the sum-of-squares hierarchy that work with roughly $m = n^{3/2}$observations, and we complement our result by showing that any attempt to solve tensor prediction with $m = n^{3/2 - \epsilon}$ observations through the sum-of-squares hierarchy needs $n^{2 \epsilon}$ rounds and consequently would run in moderately exponential time. In contrast, information theoretically roughly $m = n$ observations suffice. 
This work is part of a broader agenda of studying computational vs. statistical tradeoffs through the sum-of-squares hierarchy. In particular, for linear inverse problems (such as tensor prediction) the natural sum-of-squares relaxation gives rise to a sequence of norms. Our approach is to characterize their Rademacher complexity. Moreover, both our upper and lower bounds are based on connections between this, and the task of strongly refuting random $3$-XOR formulas, and the resolution proof system.