Institut de Mathématiques de Bourgogne,\
    Université de Bourgogne Franche-Comté,\
    9, rue Alain Savary, 21078 Dijon, France
title: Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis
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**Keywords.** Averaging, Functional data, Geometric median, Online algorithms, Online principal components, Recursive robust estimation, Stochastic gradient, Weiszfeld’s algorithm.

Introduction
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Principal Components Analysis is one of the most useful statistical tool to extract information by reducing the dimension when one has to analyze large samples of multivariate or functional data (see [*e.g.*]{} [@Jolliffe2002] or [@RamsaySilverman2005]). When both the dimension and the sample size are large, outlying observations may be difficult to detect automatically. Principal components, which are derived from the spectral analysis of the covariance matrix, can be very sensitive to outliers (see [@DGK1981]) and many robust procedures for principal components analysis have been considered in the literature (see [@HRVA2008], [@HubR2009] and [@MR2238141]).

The most popular approaches are probably the minimum covariance determinant estimator (see [@RvD99]) and the robust projection pursuit (see [@CR-G2005] and [@CFO2007]). Robust PCA based on projection pursuit has been extended to deal with functional data in [@HyndmanUllah2007] and [@BBTW2011]. Adopting another point of view, robust modifications of the covariance matrix, based on projection 