Let $X_1,..., X_n \in \mathbb{R}^d$ be independent Gaussian random vectors with independent entries and variance profile $(b_{ij})_{i \in [d],j \in [n]}$. A major question in the study of covariance estimation is to give precise control on the deviation of $\sum_{j \in [n]}X_jX_j^T-\mathbb{E} X_jX_j^T$. We show that under mild conditions, we have \begin{align*} \mathbb{E} \left\|\sum_{j \in [n]}X_jX_j^T-\mathbb{E} X_jX_j^T\right\| \lesssim \max_{i \in [d]}\left(\sum_{j \in [n]}\sum_{l \in [d]}b_{ij}^2b_{lj}^2\right)^{1/2}+\max_{j \in [n]}\sum_{i \in [d]}b_{ij}^2+\text{error}. \end{align*} The error is quantifiable, and we often capture the $4$th-moment dependency already presented in the literature for some examples. The proofs are based on the moment method and a careful analysis of the structure of the shapes that matter. We also provide examples showing improvement over the past works and matching lower bounds.

翻译：设 $X_1,..., X_n \in \mathbb{R}^d$ 为具有独立分量的独立高斯随机向量，其方差轮廓为 $(b_{ij})_{i \in [d],j \in [n]}$。协方差估计研究中的一个核心问题在于精确控制 $\sum_{j \in [n]}X_jX_j^T-\mathbb{E} X_jX_j^T$ 的偏差。我们证明，在温和条件下，有 \begin{align*} \mathbb{E} \left\|\sum_{j \in [n]}X_jX_j^T-\mathbb{E} X_jX_j^T\right\| \lesssim \max_{i \in [d]}\left(\sum_{j \in [n]}\sum_{l \in [d]}b_{ij}^2b_{lj}^2\right)^{1/2}+\max_{j \in [n]}\sum_{i \in [d]}b_{ij}^2+\text{误差项}。 \end{align*} 该误差项可量化，且在若干实例中我们精确捕捉了已有文献中出现的四阶矩依赖性。证明基于矩方法以及对关键形状结构的细致分析。此外，我们提供了实例以展示相较于以往工作的改进，并给出匹配的下界。