The metric structure of the formigram interleaving distance
Woojin Kim, Facundo Mémoli, Anastasios Stefanou · arXiv (Cornell University) · 2019
\textit{Formigrams} are a natural generalization of the notion of \textit{dendrograms}. This notion has recently been proposed as a signature for studying the evolution of clusters in dynamic datasets across different time scales. Although its formulation is set-theoretic, the notion of formigram is deeply related to certain algebraic-topological methods used in \textit{topological data analysis}, such as \textit{Reeb graphs} and \textit{zigzag persistence modules}. In this paper we give a self-contained study of the algebraic structure of formigrams and their interleaving distance. For a finite set $X$, we define a partial order on the collection of all formigrams and we show that every formigram over $X$ has a canonical decomposition into a join of simpler formigrams. This is analogous to the decomposition of persistence modules into direct sums of interval modules. Furthermore, we show that the interleaving distance between formigrams decomposes into a product metric of the interleaving distance between certain pre-cosheaves. This is analogous to the celebrated \textit{interleaving-bottleneck isometry theorem} for persistence modules.