Translation Operator in Graph Signal Processing: A Generalized Approach

TL;DR

This paper introduces a generalized translation operator for graph signals, extending classical shift concepts to irregular graph structures, and connects it to quantum dynamics via the Schrödinger equation.

Contribution

It proposes an isometric translation operator in the joint time-vertex domain and links graph translation to quantum-inspired dynamic systems.

Findings

Defined an abstract graph translation operator

Developed an isometric joint time-vertex translation operator

Connected graph translation to Schrödinger equation dynamics

Abstract

The notion of translation (shift) is straightforward in classical signal processing, however, it is challenging on an irregular graph structure. In this work, we present an approach to characterize the translation operator in various signal domains. By a natural generalization from classical domains, one can characterize an abstract representation for the graph translation operator. Then we propose an isometric translation operator in joint time-vertex domain consistent with the abstract form of translation operators in other domains. We also demonstrate the connection between this notion and the Schr\"{o}dinger equation on a dynamic system which intriguingly describes the idea behind translation on graph.