论文标题
泰勒和玛娜的M-估计剂：非反应浓度结果
Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results
论文作者
论文摘要
泰勒（Tyler's）和玛娜（Maronna）的M-估计器及其正规变体是估计多变量分布的散射或协方差矩阵的流行强大方法。在这项工作中，我们研究了这些估计值的非反应性行为，以从满足以下特性之一的分布中采样：1）独立的下高斯条目，直至线性转换； 2）对数符合分布； 3）分布满足凸浓度特性。我们的主要贡献是这些M估计器围绕适当缩放的数据样本协方差矩阵衍生了这些M估计量的紧密非反应浓度界限。在我们的工作之前，仅针对椭圆形和高斯分布而得出了非反应界限。我们的证明使用了来自非渐近随机矩阵理论和高维几何形状的各种工具。最后，我们在两个实践兴趣的例子上说明了结果的实用性：稀疏协方差和稀疏精度矩阵估计。
Tyler's and Maronna's M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: 1) independent sub-Gaussian entries, up to a linear transformation; 2) log-concave distributions; 3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, non-asymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.
