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

For an \(n\)-variate order-\(d\) tensor \(A\), define \( A_{\max} := \sup_{\| x \|_2 = 1} \langle A , x^{\otimes d} \rangle\) 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 \(\pm 1\) entries, \(A_{\max} \lesssim \sqrt{n\cdot d\cdot\log d}\) w.h.p. We study the problem of efficiently certifying upper bounds on \(A_{\max}\) 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 \(A_{\max}\) that satisfies \[ B ~~~\leq~~ A_{\max} \cdot \biggl(\frac{n}{q^{\,1-o(1)}}\biggr)^{q/4-1/2} \quad \text{w.h.p.} \] 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 \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] - When \(A\) is a random order-\(d\) tensor, we prove that \(q\) levels of SoS certifies an upper bound \(B\) on \(A_{\max}\) that satisfies \[ B ~~\leq ~~ A_{\max} \cdot \biggl(\frac{\widetilde{O}(n)}{q}\biggr)^{d/4 - 1/2} \quad \text{w.h.p.} \] 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 established the tight characterization for constant levels of SoS.