Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

TL;DR

This paper develops new scalar and matrix Chernoff bounds for distributions over the binary hypercube, based on a novel $ ext{ell}_ ext{infty}$-independence condition, extending concentration results to dependent variables.

Contribution

It introduces $ ext{ell}_ extinfty$-independence as a condition for concentration bounds, generalizing and strengthening existing matrix Chernoff bounds for dependent distributions.

Findings

Matrix Chernoff bound matches Tropp's for independent variables.

Union of $O( ext{log}|V|)$ random spanning trees yields spectral sparsifiers.

Bounds apply to broad classes of distributions over $ ext{0,1}^n$.

Abstract

We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube $\{0,1\}^n$. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is $\ell_\infty$-independent when the infinity norm of its influence matrix $\mathcal{I}$ is bounded by a constant. We show that any distribution which is $\ell_\infty$-independent satisfies a matrix Chernoff bound that matches the matrix Chernoff bound for independent random variables due to Tropp. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18). Using our bound, we can conclude as a corollary that a union of $O(\log|V|)$ random spanning trees gives a spectral graph sparsifier of a graph with $|V|$ vertices with high probability, matching results for independent edge sampling, and matching lower bounds from Kyng and Song.