Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm

Abstract

With the progress of measurement apparatus and the development of automatic sensors, it is not unusual anymore to get large samples of observations taking values in high-dimension spaces, such as functional spaces. In such large samples of high-dimensional data, outlying curves may not be uncommon, and even a few individuals may corrupt simple statistical indicators, such as the mean trajectory. We focus here on the estimation of the geometric median which is a direct generalization of the real median in metric spaces and has nice robustness properties. It is possible to estimate the geometric median, being defined as the minimizer of a simple convex functional that is differentiable everywhere when the distribution has no atom, with online gradient algorithms. Such algorithms are very fast and can deal with large samples. Furthermore, they also can be simply updated when the data arrive sequentially. We state the almost sure consistency and the $L^{2}$ rates of convergence of the stochastic gradient estimator as well as the asymptotic normality of its averaged version. We get that the asymptotic distribution of the averaged version of the algorithm is the same as the classic estimators, which are based on the minimization of the empirical loss function. The performances of our averaged sequential estimator, both in terms of computation speed and accuracy of the estimations, are evaluated with a small simulation study. Our approach is also illustrated on a sample of more than 5000 individual television audiences measured every second over a period of 24 hours.