Shift-invariant spaces on finite undirected graphs

TL;DR

This paper extends the concept of shift-invariant spaces to finite undirected graphs, introducing graph shift-invariant spaces (GSISs) and analyzing their properties, structures, and relationships with other graph signal processing frameworks.

Contribution

It introduces GSISs, characterizes their properties, and develops a new graph Fourier transform, advancing the theoretical foundation of graph signal processing.

Findings

GSISs characterized via range and fiber functions

Connection established between polynomial filters and shift-invariant filters

A graph uncertainty principle is derived

Abstract

Shift-invariant spaces (SISs) on the real line provide a natural framework for representing, analyzing and processing signals with inherent shift-invariant structure. In this paper, we extend this framework to the finite undirected graph setting by introducing the concept of graph shift-invariant spaces (GSISs). We examine several properties of GSISs, including their characterization via range functions and fiber functions in the Fourier domain, their connections to shift-invariant filters and polynomial filters, the frame and Riesz basis structures of finitely generated GSISs, and their intricate relationships with bandlimited spaces, finitely generated GSISs, and graph reproducing kernel Hilbert spaces with shift-invariant reproducing kernels (SIGRKHSs). Our analysis reveals several distinctions between SISs on the line and GSISs, such as the shift-invariance of the frame operator, the existence of shift-invariant dual frames, the emergence of fractional shift-invariance, and the interrelationships among GSISs, finitely generated GSISs, SIGRKHSs and bandlimited spaces. In this paper, we also introduce a spectral decomposition of the identity associated with graph shifts and propose a novel definition of the graph Fourier transform (GFT) of spectral type, together with explicit formulations for the GFTs on complete graphs and circulant graphs. In addition, we establish a clear connection between polynomial filters and shift-invariant filters, and we derive a graph uncertainty principle governing the essential supports of a nonzero graph signal and its GFT.