Conditioning and covariance on caterpillars

TL;DR

This paper investigates how conditioning on a small subset of binary variables can significantly reduce their pairwise covariance, with a focus on caterpillar-shaped information flow trees, relevant to correlation rounding in convex relaxations.

Contribution

It proves the conjecture that conditioning on O(1/ε) variables suffices to reduce covariance in caterpillar graphs, advancing understanding in correlation rounding for specific tree structures.

Findings

Proves the conjecture for caterpillar graphs.

Shows conditioning reduces covariance to below ε.

Provides insights for correlation rounding in convex relaxations.

Abstract

Let $X_1, \dots, X_n$ be joint $\{ \pm 1\}$-valued random variables. It is known that conditioning on a random subset of $O(1/\epsilon^2)$ of them reduces their average pairwise covariance to below $\epsilon$ (in expectation). We conjecture that $O(1/\epsilon^2)$ can be improved to $O(1/\epsilon)$. The motivation for the problem and our conjectured improvement comes from the theory of global correlation rounding for convex relaxation hierarchies. We suggest attempting the conjecture in the case that $X_1, \dots, X_n$ are the leaves of an information flow tree. We prove the conjecture in the case that the information flow tree is a caterpillar graph (similar to a two-state hidden Markov model).