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.
Statistical inference for heavy tailed series with extremal independence
本文考虑平稳时间序列$\{X_j, j \in Z\} whose finite dimensional distributions are regularly varying with extremal independence. We assume that for each $ h \geq 1 $, conditionally on $ X 0 $ to exceed a threshold tending to infinity, the conditional distribution of $ X h$适当归一化弱收敛到一个非退化分布.在本文中，我们考虑估计的正常化和极限分布。
Heavy tailed priors: an alternative to non-informative priors in the estimation of proportions on small areas
我们探索柯西和一个新的重尾（Fuquene，Perez和Pericchi（2011））先验估计小区域的比例。使用分层模型和指数族形式的二项式似然。我们相信，在调查抽样设置重尾先验可能比选择无信息先验更有效地消除对涉及主观因素或假设的方法的反感。为了说明稳健的贝叶斯方法，我们将此方法应用于一个流行的例子：“贝叶斯问题”。最后，我们建议在小区域内不存在或存在离群值时使用Cauchy先验，当离群值是特定的小区域时使用Fuquene et al.（2011）先验。
The Spurious Factor Dilemma: Robust Inference in Heavy-Tailed Elliptical Factor Models
KLASMOE and School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；Department of Statistics and Data Science, National University of Singapore（新加坡国立大学统计与数据科学系）；Department of Mathematics, Shanghai University（上海大学数学系）
Mean estimation and regression under heavy-tailed distributions--a survey
Pompeu Fabra University（庞培法布拉大学）；ICREA（加泰罗尼亚研究与高等研究院）；Barcelona Graduate School of Economics（巴塞罗那经济研究生院）；The Australian National University（澳大利亚国立大学）；Sorbonne University（索邦大学）
Privacy of SGD under Gaussian or Heavy-Tailed Noise: Guarantees without Gradient Clipping
Inria, CNRS, Ecole Normale Supérieure PSL Research University（法国国家科学研究中心、巴黎高等师范学院PSL研究大学）；Department of Management Science and Information Systems（管理科学与信息系统系）；Faculty of Engineering and Natural Sciences（工程与自然科学学院）；Department of Mathematics（数学系）
