Open Access
November 2015 Geometric median and robust estimation in Banach spaces
Stanislav Minsker
Bernoulli 21(4): 2308-2335 (November 2015). DOI: 10.3150/14-BEJ645


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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Stanislav Minsker. "Geometric median and robust estimation in Banach spaces." Bernoulli 21 (4) 2308 - 2335, November 2015.


Received: 1 November 2013; Revised: 1 May 2014; Published: November 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1348.60041
MathSciNet: MR3378468
Digital Object Identifier: 10.3150/14-BEJ645

Keywords: distributed computing , heavy-tailed noise , large deviations , linear models , low-rank matrix estimation , Principal Component Analysis , robust estimation

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 4 • November 2015
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