Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in Topological Data Analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works ~\mbox{%DIFAUXCMD Brown2019, Carriere2018a, Carriere2018, Munch2016  focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, where the domain of the filter is  in \mathbb R^d instead of \mathbb R. The only available result in this setting ~\mbox{%DIFAUXCMD Munch2016  topological spaces and cannot be used as is for finite metric spaces representing data, such as point clouds and distance matrices. 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%DIFDELCMD < {\em %%% 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 .
Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in Topological Data Analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works BBMW19, CO17, CMO18, MW16 , focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, when the codomain of the filter is \mathbb{R in this setting DMW17, MW16 can only handle continuous topological spaces and cannot be used as is for finite metric spaces representing data, such as point clouds and distance matrices. In this article, we introduce a slight modification of the usual Mapper construction %DIFDELCMD < {\em %%% }\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 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 DMW17]. We finally provide applications of this setting in statistics and machine learning for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approach 
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~\mbox{%DIFAUXCMD Brown2019, Carriere2018a, Carriere2018, Munch2016
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\mathbb R^d instead of \mathbb R. The only available result in
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~\mbox{%DIFAUXCMD Munch2016
DMW17, MW16
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can only handle continuous
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present an approximation result for the Reeb space in the multivariate case using a Mapper-based estimator, which is a
introduce a
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. Moreover, our approximation is stated with respect to a pseudometric that is an extension of the usual
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interleaving distance
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and we give risk bounds for estimating the Reeb space using this estimator. Our approach applies in particular to the setting
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the Mapper is estimated from the data. We
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
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DMW17
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