This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector $X in mathbb{R} ^{n}$ with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand [28] for Rademacher chaoses and the more recent uniform result of Adamczak [2] which holds under certain rather strong assumptions on the distribution of $X$. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici [21]. Finally, we show a uniform Hanson-Wright-type inequality in the Ising model under Dobrushin’s condition. A closely related question was posed by Marton [22].
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机译：本文致力于统一版本的Hanson-Wright Inequirality，为随机向量$ x In mathbb {r} ^ {n} $与独立的subgaussian组件。纸张的核心技术基于熵方法与兴趣函数的两个梯度的截断和$ x $本身的组件相结合。我们的结果尤其恢复了Talaghant [28]的经典统一界限，用于Rademacher Chaose和Adamczak [2]的均匀均匀效果，其在某些相当强烈的假设下持有$ x $的分布。我们提供了多种应用的技术：我们建立了标准的汉森 - 赖特不等式的版本，在一些政权中更加紧张。扩展我们的结果我们显示了一个版本的无维矩阵伯尔尼斯坦不等式，该不等式保存具有子尺寸频谱规范的随机矩阵。我们将导出的不等式应用于与缺失的观察结果的协方差估算问题，并证明了Lounici的最近结果的几乎最佳的高概率版本[21]。最后，我们在Dobrushin病情下展示了在Ising Model中的统一汉森 - 赖特型不等式。玛顿提出了密切相关的问题[22]。
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