Moments Tensors, Hilbert's Identity, and k-wise Uncorrelated Random Variables
Bo Jiang, Simai He, Zhening Li, Shuzhong Zhang · Mathematics of Operations Research · 2013
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 (xTQx)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 (xTQx)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.