  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.
