Localization bounds for the graph translation

TL;DR

This paper demonstrates that the graph translation operator can be interpreted as a diffusion process in the vertex domain, with localization properties proven through exponential decay of impulse responses, and introduces techniques for analyzing similar operators.

Contribution

It provides a vertex-domain interpretation of the graph translation operator as a diffusion process and formalizes methods for studying other graph signal operators.

Findings

Impulse response exhibits exponential decay, indicating localization preservation.

The operator can be approximated with polynomials for vertex-domain interpretation.

Techniques are formalized for analyzing other graph signal operators.

Abstract

The graph translation operator has been defined with good spectral properties in mind, and in particular with the end goal of being an isometric operator. Unfortunately, the resulting definitions do not provide good intuitions on a vertex-domain interpretation. In this paper, we show that this operator does have a vertex-domain interpretation as a diffusion operator using a polynomial approximation. We show that its impulse response exhibit an exponential decay of the energy way from the impulse, demonstrating localization preservation. Additionally, we formalize several techniques that can be used to study other graph signal operators.