Neighborhood-Preserving Translations on Graphs

TL;DR

This paper introduces a novel way to define translations on graphs that preserve neighborhood structures, aligning with traditional translations on grid graphs, and enhancing graph signal processing techniques.

Contribution

It proposes a new definition of graph translations based on neighborhood preservation, differing from existing methods, and aligns with classical translations on grid graphs.

Findings

Definitions match translations on grid graphs

Preserve neighborhood properties in graph translations

Applicable to various graph-based signal processing tasks

Abstract

In many domains (e.g. Internet of Things, neuroimaging) signals are naturally supported on graphs. These graphs usually convey information on similarity between the values taken by the signal at the corresponding vertices. An interest of using graphs is that it allows to define ad hoc operators to perform signal processing. Among them, ones of paramount importance in many tasks are translations. In this paper we are interested in defining translations on graphs using a few simple properties. Namely we propose to define translations as functions from vertices to adjacent ones, that preserve neighborhood properties of the graph. We show that our definitions, contrary to other works on the subject, match usual translations on grid graphs.