Conference abstracts

Session A4 - Computational Geometry and Topology

July 10, 17:00 ~ 17:25

Universality of the Homotopy Interleaving Distance

Michael Lesnick

Princeton University, USA   -

We introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in topological data analysis to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces $X$ and $Y$ certifies that $X$ and $Y$ are approximately isomorphic, a homotopy interleaving between $X$ and $Y$ certifies that $X$ and $Y$ are approximately weakly equivalent.

The main results of this paper are that homotopy interleavings induce an extended pseudometric $d_{HI}$ on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that $d_{HI}$ (or more generally, any pseudometric satisfying these two axioms and an additional “homology bounding” axiom) can be used to formulate lifts of fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces.

Joint work with Andrew J. Blumberg (UT Austin).

View abstract PDF

FoCM 2017, based on a nodethirtythree design.