You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
For an @math -variate order- @math tensor @math , define @math 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 @math entries, @math w.h.p. We study the problem of efficiently certifying upper bounds on @math via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: - When @math is a random order- @math tensor, we prove that @math levels of SoS certifies an upper bound @math on @math that satisfies [ B A_ ( 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 @math is large. - We show the above bound is the best possible up to lower order terms, namely the optimum of the level- @math SoS relaxation is at least [ A_ ( n q^ ,1+o(1) )^ q 4-1 2 . ] - When @math is a random order- @math tensor, we prove that @math levels of SoS certifies an upper bound @math on @math that satisfies [ B A_ ( O (n) q )^ d 4 - 1 2 w.h.p. ] For growing @math , 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.

Cited Abstract(s):
[1]: We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere.These relatively simple optimization problems have many applications.We review known approximation results as well as negative (inapproximability) results from the recent literature.
* Results for worst-case tensors. It is proved in [1] that the @math -level SoS gives an @math approximation to @math in the case of arbitrary @math -tensors and an @math approximation to @math in the case of @math -tensors with non-negative entries (for technical reasons one can only approximate @math in the former case).