论文标题
无尺寸范围，用于重型分布的依赖矩阵和操作员的总和
Dimension-free Bounds for Sum of Dependent Matrices and Operators with Heavy-Tailed Distribution
论文作者
论文摘要
我们证明了具有依赖性和{\ rc重型尾巴}的高维随机矩阵和运算符的总和。高维矩阵的估计是许多现代应用的关注点。但是，大多数结果均用于独立观察。因此，取得依赖和重尾矩阵的结果至关重要。在本文中，我们在总和的偏差上得出了无维的上限。因此，界限并不明确取决于矩阵的维度，而是取决于其有效等级。我们的结果概括了有关矩阵总和偏差的几项现有研究。它依赖于两种技术：（i）偶数生成函数的差异近似，以及（ii）通过截断矩阵的特征值来鲁棒化。我们揭示我们的结果适用于几个问题，例如协方差矩阵估计，隐藏的马尔可夫模型和过度参数化的线性回归。 一开始，我们附加了原始纸的折面。我们通过引入log-sobolev不等式代替界面条件来纠正原始论文的定理4。我们表明，在新条件下可以恢复原始论文中讨论的示例。原始纸张未校正的版本（包括上述错误）是在此透明度和比较后附加的。
We prove deviation inequalities for sums of high-dimensional random matrices and operators with dependence and {\rc heavy tails}. Estimation of high-dimensional matrices is a concern for numerous modern applications. However, most results are stated for independent observations. Therefore, it is critical to derive results for dependent and heavy-tailed matrices. In this paper, we derive a dimension-free upper bound on the deviation of the sums. Thus, the bound does not depend explicitly on the dimension of the matrices but rather on their effective rank. Our result generalizes several existing studies on the deviation of sums of matrices. It relies on two techniques: (i) a variational approximation of the dual of moment generating functions, and (ii) robustification through the truncation of the eigenvalues of the matrices. We reveal that our results are applicable to several problems, such as covariance matrix estimation, hidden Markov models, and overparameterized linear regression. At the beginning, we have attached a corrigendum of the original paper. We correct Theorem 4 of the original paper by introducing a log-Sobolev inequality in place of the boundedness condition. We show that the examples discussed in the original paper can be recovered under new conditions. The original paper, uncorrected version -- which includes the aforementioned error -- is appended after this corrigendum for transparency and comparison.
