arXiv subjects

Živa Urbančič

Publications and source records attributed to Živa Urbančič.

2 recordsLinked to original sources

arXiv2025

Gromov-Hausdorff Distance for Directed Spaces

The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The directed Gromov-Hausdorff distance measures the distance between two extended metric spaces, where the new metric, defined on the same underlying space, is induced by the length of zigzag paths. This distance is then computed by isometrically embedding the directed metric spaces into an ambient directed space equipped with the zigzag distance. Analogously to the classical Gromov-Hausdorff distance, we also propose alternative formulations based on the distortion of d-maps and d-correspondences. However, unlike the classical case, these directed distances are not equivalent.

math.AT↗

arXiv2023

Ladder Decomposition for Morphisms of Persistence Modules

The output of persistent homology is an algebraic object called a persistence module. This object admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. In this paper we investigate when a morphism $Φ\colon V \to W$ of persistence modules admits an analogous direct sum decomposition. Jacquard et al. showed that a ladder decomposition can be obtained whenever the barcodes of $V$ and $W$ do not have any strictly nested bars. We refine this result and show that even in the presence of nested bars, a ladder decomposition exists when the morphism is sufficiently close to being invertible relative to the scale of the nested bars.

math.AT↗