Assignments to sheaves of pseudometric spaces

TL;DR

This paper introduces the concept of the consistency filtration for sheaves of pseudometric spaces, demonstrating its continuity, functoriality, and robustness to noise, which aids in analyzing local-to-global data consistency.

Contribution

It formalizes the consistency filtration as a functor and proves its robustness, advancing the theoretical framework for sheaf-based data analysis.

Findings

Consistency radius is a continuous map.

Consistency filtration forms a structure-preserving functor.

Robustness to noisy data is established.

Abstract

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which quantifies the agreement between overlapping local sections in the assignment -- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.