Scalar and Matrix Chernoff Bounds from ℓ_∞-Independence

Abstract

We present new scalar and matrix Cherno↵-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 `1-independent when the infinity norm of its influence matrix I is bounded by a constant. We show that any distribution which is `1-infinity independent satisfies a matrix Cherno↵ bound that matches the matrix Cherno↵ bound for independent random variables due to Tropp. Our matrix Cherno↵ bound is a broad generalization and strengthening of the matrix Cherno↵ 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.