4.4 Article

Geometric median and robust estimation in Banach spaces

Journal

BERNOULLI
Volume 21, Issue 4, Pages 2308-2335

Publisher

INT STATISTICAL INST
DOI: 10.3150/14-BEJ645

Keywords

distributed computing; heavy-tailed noise; large deviations; linear models; low-rank matrix estimation; principal component analysis; robust estimation

Funding

  1. National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH) [NSF DMS-0847388, NSF CCF-0808847, R01-ES-017436]

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In many real-world applications, collected data are contaminated by noise with heavy-tailed distribution and might contain outliers of large magnitude. In this situation, it is necessary to apply methods which produce reliable outcomes even if the input contains corrupted measurements. We describe a general method which allows one to obtain estimators with tight concentration around the true parameter of interest taking values in a Banach space. Suggested construction relies on the fact that the geometric median of a collection of independent weakly concentrated estimators satisfies a much stronger deviation bound than each individual element in the collection. Our approach is illustrated through several examples, including sparse linear regression and low-rank matrix recovery problems.

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