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. We ] provide applications of this setting in statistics and machine learning and probability for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approach .
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 } where the filter function used to compute Mapper is also estimated from data, such as the eigenfunctions of PCA. Our results are given with respect to the Gromov-Hausdorff distance, computed with specific filter-based pseudometrics for Mappers and Reeb spaces defined in ] provide applications of this setting in statistics and machine learning and probability for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approach .