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 $\ell_\infty$-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.

翻译：我们为研究离散分布的Markov链条混合时间而开发了最近工具,我们说,当其影响矩阵无限性规范$\mathcal{I}受一个常数的约束时,分配是独立的。我们显示,任何分布值为$@infty$-infinity的独立独立,都符合Chernoff矩阵,该矩阵与Tropp应具备的独立随机变量相匹配。我们的Chernoff矩阵是广泛的,加强了Kyng和Song(FOCS'18)的Chernoff矩阵。我们可以用我们的约束,得出一个推论,即由$O(log_V}}$)的随机覆盖树木结合,以高概率的$+ ⁇ V$-infty$-infinfinity为图谱,匹配独立边缘取样结果,并匹配Ky和Song的较低约束值。