Vector-valued Graph Signal Processing
Antonio Caputo
TL;DR
The paper extends graph signal processing to $X$-valued signals on a finite graph $G$, representing signals as $X^N$ or Bochner spaces $L^p(\mathcal{V}(G), X)$. It develops a vector-valued graph Fourier transform using an orthonormal basis $\mathcal{B}$ of $\mathbb{K}^N$ and proves that the transform yields a topological isomorphism $L^p(\mathcal{V}(G), X) \to L^q(\mathcal{V}(G), X)$ with bounds depending on $N$ and the matrix coherence $\|U\|_\infty$; in the Hilbert case, Parseval/Plancherel give unitary behavior on $L^2$. The paper also defines convolution and translation for $X$-valued signals, showing $\widehat{\alpha*x}=\hat{\alpha}\hat{x}$, establishing bilinearity, commutativity, associativity, and a Young-type inequality, and formulates a vector-valued primary uncertainty principle with lower bound $(1/\|U\|_\infty)^2$. These results enable time-frequency analysis on graphs for multi-signal data and provide a solid functional-analytic framework for vector-valued graph signal processing.
Abstract
Classical graph signal processing (GSP) introduces methodologies for analyzing real or complex signals defined on graph domains, moving beyond classical uniform sampling techniques, such as the graph discrete Fourier transform (GDFT), employed as a pivotal tool for transforming graph signals into their spectral representation, enabling effective signal processing techniques such as filtering and denoising. In this paper, we propose a possible generalization of the set of signals and we study some properties of the more general set of vector-valued signals, which take values into any Banach space, and some properties of the fundamental operators of vertex-frequency analysis acting on these signals, such as the Fourier transform, the convolution operator and the translation operator. In particular, we show some estimates involving their operator norm as linear operators between Banach spaces and we establish a graph version of the classical primary uncertainty principle. We also show how these estimates depend on the choice of an orthonormal basis of $\mathbb{K}^N$. The importance of considering this general set of signals derives from the possibility to study multiple signals at the same time and the correlation existing between them, since multiple scalar signals can be modelled as a unique vector-valued signal.