Robust covariance and scatter matrix estimation under Huber’s contamination model

By Mengjie Chen, Chao Gao, Zhao Ren. 2015

201 citations recorded by the source corpus.

Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber's $\epsilon$-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.

Subfields: Computational Mechanics, Statistics and Probability

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Citations in the corpus

Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates (2018, methodology): “This proof is essentially the same as the lower bound in the robust mean estimation literature (Chen et al., 2015, Lai et al., 2016).”
High‐dimensional covariance matrix estimation (2019, background): “Chen et al. (2018) showed that in fact, for any U 2 S p (cid:0) 1 , D U ( (cid:12) (cid:6) ; P (cid:6) ) = 1 = 2, where (cid:12) is such that (cid:8)( p (cid:12) ) = 3 = 4.”
Dataset Security for Machine Learning: Data Poisoning, Backdoor Attacks, and Defenses (2020, background): “A significant amount of work in this space shows that, from an information-theoretic point of view, this problem is indeed tractable for a wide variety of tasks and data distributions [157], [159], [160], [161], [162], [163].”
Robust multivariate mean estimation: The optimality of trimmed mean (2019, methodology): “Chen, Gao, and Ren [3] develop a general theory of minimax bounds under Huber’s contamination model (i.e., when the contamination is i.i.d.) for parametric families of distributions.”
High-Dimensional Robust Mean Estimation in Nearly-Linear Time (2018, background): “For example, the Tukey median [Tuk75] is a sample-eﬃcient robust mean estimator for various symmetric distributions [DG92, CGR15].”
Adaptive Huber Regression (2017, background): “We refer to Chen, Gao, and Ren (2018) for the most recent result and a literature review concerning this problem.”