We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples. Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.

翻译：我们提出了一种获取随机矩阵范数界的新方法，其中每个矩阵元是底层独立实值随机变量的低次多项式。此类矩阵出现在谱分析与优化算法的多种场景中，这些场景需要理解依赖于独立样本数据的随机矩阵谱。通过运用分析中的解耦与线性化思想，我们展示了一种用对应于导数的低次多项式矩阵来表达此类矩阵范数界的简明方法。迭代应用该方法可得到一个具有初等证明的简洁界，该界能够复现许多先前需要更复杂技术才能得到的估计结果。