论文标题
强大瑞利分布的矩阵伯恩斯坦矩阵不平等
A Matrix Bernstein Inequality for Strong Rayleigh Distributions
论文作者
论文摘要
熵方法通过建立诸如庞卡雷和log-sobolev不平等等功能不平等的功能不平等，为证明标量浓度不平等的不等式提供了强大的框架。这些不平等对于来自马尔可夫链的固定分布的依赖分布而得出浓度特别有用。与标量案例相反，对基质浓度不平等的研究是相当近期的，并且似乎没有使用熵方法证明基质浓度不平等的一般框架。在本文中，我们启动了矩阵有价值的功能不平等的研究，并展示了如何使用矩阵有价值的功能不平等来建立一种迭代方差比较论点，以证明基质伯恩斯坦（Matrix Bernstein）如不平等。强瑞利分布是一类分布，可满足强大的负依赖性特性。我们表明，在Hermon和Salez [HS19]工作中考虑的Flip-Swap Markov连锁店最近确定的标量庞加罗不平等现象可以证明是矩阵有价值的庞加利不平等现象。然后，可以将其与迭代方差参数结合使用，以证明用于Lipshitz Matrix-varued函数的矩阵Bernstein不平等版本，用于强瑞利度量。
The Entropy method provides a powerful framework for proving scalar concentration inequalities by establishing functional inequalities like Poincare and log-Sobolev inequalities. These inequalities are especially useful for deriving concentration for dependent distributions coming from stationary distributions of Markov chains. In contrast to the scalar case, the study of matrix concentration inequalities is fairly recent and there does not seem to be a general framework for proving matrix concentration inequalities using the Entropy method. In this paper, we initiate the study of Matrix valued functional inequalities and show how matrix valued functional inequalities can be used to establish a kind of iterative variance comparison argument to prove Matrix Bernstein like inequalities. Strong Rayleigh distributions are a class of distributions which satisfy a strong form of negative dependence properties. We show that scalar Poincare inequalities recently established for a flip-swap markov chain considered in the work of Hermon and Salez [HS19] can be lifted to prove matrix valued Poincare inequalities. This can then be combined with the iterative variance argument to prove a version of matrix Bernstein inequality for Lipshitz matrix-valued functions for Strong Rayleigh measures.
