Before revision:  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:  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 } and we give risk bounds for estimating the Reeb space using this estimator. Our approach applies in particular to the setting where the filter function used to compute the Mapper is estimated from the data.