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Zhao Ren University of Pittsburgh

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Robust Covariance/Scatter Matrix Estimation via Matrix Depth
08 September 2016 from 4:00 PM to 5:00 PM
201 Thomas Bldg
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Zhao Ren

Covariance matrix estimation is one of the most important problems in
statistics. To deal with modern complex data sets, not only do we need
estimation procedures to take advantage of the structural assumptions of
the covariance matrix, it is also important to design methods that are
resistant to arbitrary source of outliers. In this paper, we define a new
concept called matrix depth and propose to maximize the empirical matrix depth function to obtain a robust covariance matrix estimator. The
proposed estimator is shown to achieve minimax optimal rate under Huber's $\epsilon$-contamination model for estimating covariance/scatter matrices with various structures such as bandedness and sparsity.

In the second part of this talk, following the above framework, we further
establish a general decision theory for robust statistics under Huber's
$\epsilon$-contamination model. We propose a solution using Scheff{\'e}
estimate to a robust two-point testing problem that leads to the
construction of robust estimators adaptive to the proportion of
contamination. Applying the general theory, we construct robust estimators
for nonparametric density estimation, sparse linear regression and
low-rank trace regression. We show that these new estimators achieve the
minimax rate with optimal dependence on the contamination proportion. This
testing procedure, Scheff{\'e} estimate, also enjoys an optimal rate in
the exponent of the testing error, which may be of independent interest.

This is a joint work with Mengjie Chen and Chao Gao.





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