Interleaving by parts for persistence in a poset
Woojin Kim, Facundo Mémoli, Anastasios Stefanou · arXiv (Cornell University) · 2019
Metrics in computational topology are often either (i) themselves in the form of the interleaving distance $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})$ between certain order-preserving maps $\mathbf{F},\mathbf{G}:(\mathcal{P},\leq)\rightarrow (\mathcal{Q},\leq)$ between posets or (ii) admit $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})$ as a tractable lower bound, where the domain poset $(\mathcal{P},\leq)$ is equipped with a flow. In this paper, assuming that $\mathcal{Q}$ admits a join-dense subset $B$, we propose certain join representations $\mathbf{F}=\bigvee_{b\in B} \mathbf{F}_b$ and $\mathbf{G}=\bigvee_{b\in B} \mathbf{G}_b$ which satisfy $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})=\bigvee_{b\in B} d_{\mathrm{I}}(\mathbf{F}_b,\mathbf{G}_b)$ where each $d_{\mathrm{I}}(\mathbf{F}_b,\mathbf{G}_b)$ is relatively easy to compute. We leverage this result in order to (i) elucidate the structure and computational complexity of the interleaving distance for poset-indexed clusterings (i.e. poset-indexed subpartition-valued functors), (ii) to clarify the relationship between the erosion distance by Patel and the graded rank function by Betthauser, Bubenik, and Edwards, and (iii) to reformulate and generalize the tripod distance by the second author.