A neighborhood-preserving translation operator on graphs

TL;DR

This paper introduces neighborhood-preserving translation operators on graphs that mimic Euclidean translations, work on any graph, and are useful for graph signal processing, despite being NP-Complete to identify exactly.

Contribution

The paper proposes a novel neighborhood-preserving translation operator on graphs that directly operates in the vertex domain, unlike spectral methods, and demonstrates their properties and approximations.

Findings

Operators match Euclidean translations on grid graphs.

Identifying exact translations is NP-Complete.

Relaxed operators enable practical translation on arbitrary graphs.

Abstract

In this paper, we introduce translation operators on graphs. Contrary to spectrally-defined translations in the framework of graph signal processing, our operators mimic neighborhood-preserving properties of translation operators defined in Euclidean spaces directly in the vertex domain, and therefore do not deform a signal as it is translated. We show that in the case of grid graphs built on top of a metric space, these operators exactly match underlying Euclidean translations, suggesting that they completely leverage the underlying metric. More generally, these translations are defined on any graph, and can therefore be used to process signals on those graphs. We show that identifying proposed translations is in general an NP-Complete problem. To cope with this issue, we introduce relaxed versions of these operators, and illustrate translation of signals on random graphs.