Geometric Median and Robust Estimation in Banach Spaces

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 estimation of the mean and covariance matrix of a heavy-tailed distribution in Euclidean space, sparse linear regression and low-rank matrix recovery problems.