Interleaving distances from height-difference functions on posets

TL;DR

This paper introduces a new categorical interleaving distance based on height-difference functions for functor categories over arbitrary posets, generalizing classical interleaving distances and establishing stability properties.

Contribution

It develops a novel interleaving distance framework for arbitrary posets using height-difference functions, extending classical methods to more general settings.

Findings

Defines the height-interleaving distance $d_{\rho}$ for functor categories.

Shows $d_{\rho}$ coincides with classical interleaving distance on $\mathbb{R}^d$.

Proves stability of $d_{\rho}$ under perturbations of height functions.

Abstract

Interleaving distances provide a fundamental tool for comparing persistence modules and have been widely used in topological data analysis. Their definitions are typically based on translation structures (shift operations) on the indexing poset, but on general posets such structures can be scarce, making this framework restrictive. In this paper, we introduce a new interleaving-type distance for functor categories over arbitrary posets, induced by a height-difference function $\rho$. The key idea is to associate to $\rho$ an $\mathbb{R}_{\geq 0}$-indexed family of adjoint endofunctors on $\mathrm{Fun}(P,\mathcal{C})$, which play the role of generalized translations and allow us to formulate interleavings in a purely categorical manner and define the distance $d_{\rho}$, called the height-interleaving distance. In particular, any height function (i.e., a real-valued order-preserving map) canonically induces such a height-difference function, so our framework remains useful on finite posets. Moreover, when $P=\mathbb{R}^d$ and $\rho=\rho_{\mathrm{diag}}$, the resulting distance coincides with the classical interleaving distance for multiparameter persistence modules. However, in general, $d_{\rho}$ need not satisfy the triangle inequality. Under suitable hypotheses (e.g. CIP for $(P,\rho)$; this condition is automatic when $P$ is a tree poset), we prove a triangle inequality up to an additive defect bounded by a constant $c(\rho)$; in particular, when $c(\rho)=0$ this yields an extended pseudo-distance. We also establish a stability property with respect to perturbations of height-difference functions: small changes in $\rho$ induce small changes in the associated height-interleaving distance. Finally, we study an analogue of the erosion-type constructions from classical interleavings within our framework.