Frames and Bases of Translates of Signals on Undirected Graphs

TL;DR

This paper characterizes when systems of translated signals on undirected graphs form orthonormal bases, linearly independent sets, or frames, extending classical signal processing concepts to graph signals with applications to spectral graph wavelets.

Contribution

It provides new theoretical characterizations for translates on graphs to form bases, independent sets, and frames, including conditions for multiple generators and spectral graph wavelet systems.

Findings

Characterization of orthonormal bases of translated signals on graphs.

Necessary and sufficient conditions for linear independence and orthonormality.

Conditions for systems to form frames using multiple generators.

Abstract

We study a shift invariant space on an undirected graphs $G$ having $N$ vertices. We obtain a characterization theorem for a system of generalized translates $\{T_{i}g : 1\leq i\leq N\}$, for $g\in C^N$, to form an orthonormal basis. Moreover, we find a necessary and sufficient condition for the system $\{T_{i}g : 1\leq i\leq m\}$, $m\leq N$, to form a linearly independent set and an orthonormal set. Further, we obtain a characterization result for a system of generalized translates which is generated by multiple generators $g_{1},...,g_{M}$ to form a frame for $C^N$. In particular, we deduce similar results for the system $\{T_{i}M_{s}g : 1\leq i,s\leq N\}$ with modulation $M_{s}$ and the spectral graph wavelet system. We also provide an illustration for the spectral graph wavelet system.