Interleaving by parts: join decompositions of interleavings and join-assemblage of geodesics (Q6615247)

Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distance \(d_{\mathrm{I}}\)\ between poset maps. This paper proposes a representation of a poset map\N\[\N\boldsymbol{F}:\mathcal{P}\rightarrow\mathcal{Q}\N\]\Nas a join \(\vee_{b\in B}\boldsymbol{F}_{b}\)\ of simpler poset maps \(\boldsymbol{F}_{b}\),\ which in turn yields a decomposition of \(d_{\mathrm{I}}\)\ into a product metric. Scoccola [\url{https://ir.lib.uwo.ca/etd/7119/}] introduced the notion of a locally persistent category, which enables the authors to define an interleaving distance between objects in the category with a notion of approximate morphism, encompassing many distances in TDA and facilitating a uniform treatment of those distances.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] reviews the notions of lattices, subpartitions, interleaving distances, and formigrams.\N\N\item[\S 3] addresses the following results.\N\N\begin{itemize}\N\item[(1)] The authors identify join representations \(\boldsymbol{F}=\vee_{b\in B}\boldsymbol{F}_{b}\)\ and \(\boldsymbol{G}=\vee_{b\in B}\boldsymbol{G}_{b}\)\ such that\N\[\Nd_{\mathrm{I}}\left( \boldsymbol{F},\boldsymbol{G}\right) =\sup_{b\in B}\,d_{\mathrm{I}}\left( \boldsymbol{F}_{b},\boldsymbol{G}_{b}\right)\N\]\Nwhere each of the \(\boldsymbol{F}_{b}s\)\ and \(\boldsymbol{G}_{b}s\)\ is structurally simple (Theorem 3.7).\N\N\item[(2)] It is shown (Theorem 3.10) that \ is an \(\mathit{l}^{\infty}\)-product of multiple copies of a distance between upper sets in \(\mathcal{P}\).\N\N\item[(3)] It is shown (Theorem 3.13) that a distance between poset maps \(\boldsymbol{F},\boldsymbol{G}:\mathcal{P}\rightarrow\mathcal{Q}\)\ is geodesic under the assumption that\(\mathcal{Q}\) \ is a complete lattice.\N\end{itemize}\N\N\item[\S 4] addresses the following results.\N\N\begin{itemize}\N\item[(1)] It is shown (Theorem 4.3) that computing the erosion distance [\textit{A. Patel}, J. Appl. Comput. Topol. 1, No. 3--4, 397--419 (2018; Zbl 1398.18015)] between rank functions of persistence modules amounts to computing a finite number of Hausdorff distances between certain geometric signatures of graded rank functions [\textit{L. Betthauser} et al., Discrete Comput. Geom. 67, No. 1, 203--230 (2022; Zbl 1544.55003)].\N\N\item[(2)] It is shown (Theorem 4.11) that the distance, as a generalization of the Gromov-Hausdorff distance between metric spaces to one between simplicial filtrations over \(\mathcal{P}\),\ inherits the universality property of the original Gromov-Hausdorff distance, of which the celebrated Vietoris-Rips filtration stability theorem [\textit{S. Chowdhury} and \textit{F. Mémoli}, Inf. Inference 8, No. 4, 757--787 (2019; Zbl 1471.62387); \textit{F. Chazal} et al., Geom. Dedicata 173, 193--214 (2014; Zbl 1320.55003)] becomes a consequence.\N\N\item[(3)] The equivalence between several known metrics on multiparameter hierarchical clusterings is established (Theorem 4.24).\N\N\item[(4)] The computational complexity of the interleaving distances between formigrams is elucidated (Theorem 4.30).\N\end{itemize}\N\N\item[\S 5] discusses open problems.\N\end{itemize}