Statistics and Data Analysis Seminar

SPEAKER: Anastasios Stefanou

AFFILIATION: Ohio State University

TITLE: Interleaving by Parts for Persistence in a Poset.

ABSTRACT: Metrics in computational topology are often either (i) themselves in the form of the interleaving distance $d_I(F,G)$ between certain order-preserving maps $F,G:P\to Q$ between posets or (ii) admit $d_I(F,G)$ as a tractable lower bound, where the domain poset $P$ is equipped with a flow. In this paper, assuming that $Q$ admits a join-dense subset $B$, we propose certain join representations $F=\bigvee_{b\in B} F_b$ and $G=\bigvee_{b\in B} G_b$ which satisfy $d_I(F,G)=\bigvee_{b\in B} d_I(F_b,G_b)$ where each $d_I(F_b,G_b)$ 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 generalise the tripod distance by the second author.  This is a join work with Woojin Kim and Facundo Mémoli.