The Companion Model -- a Canonical Model in Graph Signal Processing

TL;DR

This paper introduces a canonical graph signal processing model using the companion graph and shift, transforming any GSP model into this form via the graph z-transform, and shows that convolution can be efficiently computed using FFT.

Contribution

It defines a canonical GSP model with a companion graph and shift, and introduces the graph z-transform to convert any GSP model into this form.

Findings

Any GSP model can be transformed into the canonical model.

The canonical model closely resembles discrete signal processing structures.

Convolution in the canonical model is efficiently computed using FFT.

Abstract

This paper introduces a $\textit{canonical}$ graph signal model defined by a $\textit{canonical}$ graph and a $\textit{canonical}$ shift, the $\textit{companion}$ graph and the $\textit{companion}$ shift. These are canonical because, under standard conditions, we show that any graph signal processing (GSP) model can be transformed into the canonical model. The transform that obtains this is the graph $z$-transform ($\textrm{G$z$T}$) that we introduce. The GSP canonical model comes closest to the discrete signal processing (DSP) time signal models: the structure of the companion shift decomposes into a line shift and a signal continuation just like the DSP shift and the GSP canonical graph is a directed line graph with a terminal condition reflecting the signal continuation condition. We further show that, surprisingly, in the canonical model, convolution of graph signals is fast convolution by the DSP FFT.