Fourier transform of vector-valued graph signals

TL;DR

This paper extends classical graph signal processing to vector-valued signals in Banach spaces, defining new spectral operators and uncertainty principles to analyze multivariate signals on complex networks.

Contribution

It introduces a generalized GSP framework for vector-valued signals, including new operators and uncertainty principles, broadening analysis capabilities for multivariate data.

Findings

Derived operator norm estimates for Fourier, convolution, translation operators.

Established graph-theoretic uncertainty principles for vector-valued signals.

Demonstrated modeling of multiple signals as a single vector entity enhances analysis.

Abstract

Classical Graph Signal Processing (GSP) provides a robust framework for analyzing signals on irregular domains, utilizing the graph Fourier transform as a cornerstone for spectral analysis and filtering. However, as data structures grow in complexity, there is an increasing need to handle multi-dimensional information. In this paper, we propose a generalization of the GSP framework by introducing vector-valued graph signals which take values in arbitrary Banach spaces. We define and investigate the fundamental operators of vertex-frequency analysis within this broader setting, including the Fourier transform, convolution, and translation operators. A key contribution of this work is the derivation of operator norm estimates and the establishment of graph-theoretic versions of classical uncertainty principles. We demonstrate how these results depend on the choice of the orthonormal basis and on the underlying $L^p$ norms. By modeling multiple scalar signals as a single vector-valued entity, this framework facilitates the study of inter-signal correlations, providing a flexible and mathematically grounded environment for analyzing multivariate time-series and time-varying signals on complex networks.