A Relative Theory of Interleavings

TL;DR

This paper develops a generalized theory of interleavings for posets in topological data analysis, enabling analysis of structures like zig-zag posets and face relation posets through mapping to more tractable posets.

Contribution

It introduces a novel approach to define interleavings via mappings to posets with translations, and provides a discretization method for interleaving inference in complex poset modules.

Findings

Provides a framework for interleaving in non-translation posets.

Introduces pixelization techniques for approximating interleavings.

Applies theory to cosheaves over metric spaces and grid structures.

Abstract

The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset have a superlinear family of translations or a similar structure. However, many posets of interest to topological data analysis, such as zig-zag posets and the face relation poset of a cell-complex, do not admit interesting translations, and consequently don't admit a nice theory of interleavings. In this paper we show how one can side-step this limitation by providing a general theory where one maps to a poset that does admit interesting translations, such as the lattice of down sets, and then defines interleavings relative to this map. Part of our theory includes a rigorous notion of discretization or "pixelization" of poset modules, which in turn we use for interleaving inference. We provide an approximation condition that in the setting of lattices gives rise to two possible pixelizations, both of which are guaranteed to be close in the interleaving distance. Finally, we conclude by considering interleaving inference for cosheaves over a metric space and give an explicit description of interleavings over a grid structure on Euclidean space.