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

With the progress of measurement apparatus and the development of automatic it is not unusual anymore to get thousands of samples of observations values in high dimension spaces such as functional spaces. In such large of high dimensional data, outlying curves may not be uncommon and even few individuals may corrupt simple statistical indicators such as the mean. We focus here on the estimation of the geometric median which is a generalization of the real median and has nice robustness properties. geometric median being defined as the minimizer of a simple convex that is differentiable everywhere when the distribution has no, it is possible to estimate it with online gradient algorithms. Such are very fast and can deal with large samples. Furthermore they also be simply updated when the data arrive sequentially. We state the almost consistency and the L2 rates of convergence of the stochastic gradient as well as the asymptotic normality of its averaged version. We get the asymptotic distribution of the averaged version of the algorithm is same as the classic estimators which are based on the minimization of the loss function. The performances of our averaged sequential estimator, in terms of computation speed and accuracy of the estimations, are with a small simulation study. Our approach is also illustrated on a of more 5000 individual television audiences measured every second over period of 24 hours.