Faithful sets of topological descriptors and the algebraic K-theory of multi-parameter zig-zag grid persistence modules

Abstract

Given a geometric simplicial complex, the uncountable set of (augmented) persistence diagrams corresponding to lower-star filtrations taken with respect to all possible directions uniquely correspond to the simplicial complex, i.e., the set is faithful. While this hints towards interesting applications in shape comparison, the set of all possible directions is uncountably infinite, and so has no hope of computability. In practice, one might use a finite approximation, but faithfulness of this approximation is not guaranteed. Motivated by the need for both computability and provable faithfulness, we provide an explicit description of a finite faithful set of augmented persistence diagrams. We then show this construction applies to augmented Euler characteristic curves and augmented Betti curves, and is stable under particular perturbations. In the specific case where the underlying complex is a graph, we provide an improved construction that utilizes a radial binary search. We then shift focus to comparing the cardinalities of minimal faithful sets of descriptors as a way to define and order equivalence classes of topological descriptor types. Focusing on six topological descriptor types commonly used in practice, we give a partial order on their corresponding equivalence classes, as well as give bounds on the sizes of minimum faithful sets for each descriptor type. Next, we broaden our view to zig-zag grid persistence modules, functors whose domain categories are posets with grid-like structure. We begin by explicitly defining such persistence modules in terms of constructible cosheaves over stratified Euclidean space, including a careful treatment of augmented persistence modules, which are analogous to the aforementioned augmented descriptors who played a central role in discussion of faithful sets. Exodromy gives us an equivalence between persistence modules as a functor category and as constructible cosheaves; we furthermore show the equivalence of these categories with a category of constructible functors out of Rd with a fixed stratification and localized at weak equivalences, essentially "standardizing" modules so that the category has a clear monoid structure. We compute the algebraic K-theory of zig-zag grid persistence modules, using a double inductive argument to show the K-theory is additive over strata. Finally, we identify connections to related topics, such as the virtual diagrams of Bubenik and Elchensen, as well as Euler characteristic and Betti curves/surfaces/manifolds. We hope a study of K-groups will provide interesting insights into the nature of persistence modules, and we indicate ways in which the zeroth and first K-groups may be interpreted