Scalar and Matrix Chernoff Bounds from ℓ∞-Independence

We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube 0, 1n. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is ℓ∞-independent when the infinity norm of its influence matrix is bounded by a constant. We show that any distribution which is ℓ∞-infinity 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.