Moments tensors, Hilbert's identity, and k-wise uncorrelated random variables

In this paper, we introduce a notion to be called k-wise uncorrelated random variables, which is similar but not identical to the so-called k-wise independent random variables in the literature. We show how to construct k-wise uncorrelated random variables by a simple procedure. The constructed random variables can be applied, e.g., to express the quartic polynomial (x^TQx)², where Q is an n×n positive semidefinite matrix, by a sum of fourth powered polynomial terms, known as Hilbert's identity. By virtue of the proposed construction, the number of required terms is no more than 2n⁴+n. This implies that it is possible to find a (2n⁴+n)-point distribution whose fourth moments tensor is exactly the symmetrization of Q⊗Q. Moreover, we prove that the number of required fourth powered polynomial terms to express (x^TQx)² is at least n(n+1)/2. The result is applied to prove that computing the matrix 2↦4 norm is NP-hard. Extensions of the results to complex random variables are discussed as well.