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

TL;DR

This paper proves NP-hardness results for the problem of finding signings of vectors that minimize the covariance operator norm, which is relevant in discrepancy theory and randomized experiments.

Contribution

It establishes the computational hardness of approximating the minimal covariance operator norm for signed sums of vectors under various expectation constraints.

Findings

NP-hardness of distinguishing zero and large covariance norm cases

Hardness persists for vectors with expectations near any fixed p

Results imply no efficient algorithms can approximate the problem within certain bounds

Abstract

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 $\Omega(1)$. 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 $\Omega((1-|p|)^2)$.