Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere

For an n-variate order-d tensor A, define Amax := sup‖x‖2=1〈A, x ⊗d〉 to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d. ±1 entries, Amax . √ n · d · log d w.h.p. We study the problem of efficiently certifying upper bounds on Amax via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: When A is a random order-q tensor, we prove that q levels of SoS certifies an upper bound B on Amax that satisfies B ≤ Amax · ( n q 1−o(1) )q/4−1/2 w.h.p. Our upper bound improves a result of Montanari and Richard (NIPS 2014) when q is large. We show the above bound is the best possible up to lower order terms, namely the optimum of the level-q SoS relaxation is at least Amax · ( n q 1+o(1) )q/4−1/2 . When A is a random order-d tensor, we prove that q levels of SoS certifies an upper bound B on Amax that satisfies B ≤ Amax · ( Õ(n) q )d/4−1/2 w.h.p. For growing q, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who gave the tight characterization for constant levels of SoS. 1998 ACM Subject Classification G.1.6 Optimization, F.2.1 Numerical Algorithms and Problems