The geometric median covariation matrix is a robust multivariate indicator
of dispersion which can be extended to infinite dimensional spaces. We define estima-
tors, based on recursive algorithms, that can be simply updated at each new observation
and are able to deal rapidly with large samples of high-dimensional data without being
obliged to store all the data in memory. Asymptotic convergence properties of the
recursive algorithms are studied under weak conditions in general separable Hilbert
spaces. The computation of the principal components can also be performed online
and this approach can be useful for online outlier detection. A simulation study clearly
shows that this robust indicator is a competitive alternative to minimum covariance
determinant when the dimension of the data is small and robust principal components
analysis based on projection pursuit and spherical projections for high-dimension data.
An illustration on a large sample and high-dimensional dataset consisting of individual
TV audiences measured at a minute scale over a period of 24 h confirms the interest of
considering the robust principal components analysis based on the median covariation
matrix. All studied algorithms are available in the R package Gmedian on CRAN.

• Fast estimation of the median covariation matrix with application to online robust principal components analysis