Before revision:  In this article, we present an approximation result for the Reeb space in the multivariate case using a Mapper-based estimator, which is a slight modification of the usual Mapper construction . Moreover, our approximation is stated with respect to a pseudometric that is an extension of the usual interleaving distance between persistence modules~\mbox{%DIFAUXCMD \cite{Chazal2016}\hspace{0pt}%DIFAUXCMD . Finally, we apply our results to the case }  where the filter function used to compute the Mapper is estimated from the data.
After revision:  In this article, we present an approximation result for the Reeb space in the multivariate case using a Mapper-based estimator, which is a slight modification of the usual Mapper construction  interleaving distance between persistence modules~\mbox{%DIFAUXCMD \cite{Chazal2016}\hspace{0pt}%DIFAUXCMD . Finally, we apply our results to the case } where the filter function used to compute the Mapper is estimated from the data.