Hardness Results for Minimizing the Covariance of Randomly Signed Sum of Vectors

Given vectors $\mathbb{v}_1, \ldots, \mathbb{v}_n \in \mathbb{R}^d$ with Euclidean norm at most $1$ and $\mathbb{x}_0 \in [-1,1]^n$, our goal is to sample a random signing $\mathbb{x} \in \{\pm 1\}^n$ with $\mathbb{E}[\mathbb{x}] = \mathbb{x}_0$ such that the operator norm of the covariance of the signed sum of the vectors $\sum_{i=1}^n \mathbb{x}(i) \mathbb{v}_i$ is as small as possible. This problem arises from the algorithmic discrepancy theory and its application in the design of randomized experiments. It is known that one can sample a random signing with expectation $\mathbb{x}_0$ and the covariance operator norm at most $1$. In this paper, we prove two hardness results for this problem. First, we show it is NP-hard to distinguish a list of vectors for which there exists a random signing with expectation ${\bf 0}$ such that the operator norm is $0$ from those for which any signing with expectation ${\bf 0}$ must have the operator norm $Ω(1)$. Second, we consider $\mathbb{x}_0 \in [-1,1]^n$ whose entries are all around an arbitrarily fixed $p \in [-1,1]$. We show it is NP-hard to distinguish a list of vectors for which there exists a random signing with expectation $\mathbb{x}_0$ such that the operator norm is $0$ from those for which any signing with expectation ${\bf 0}$ must have the operator norm $Ω((1-|p|)^2)$.