Graph signal interpolation with positive definite graph basis functions

• Published in: Applied and computational harmonic analysis Vol. 60; pp. 368 - 395
• Main Author: Erb, Wolfgang
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2022
• Subjects: Approximation errors for interpolation; Graph basis functions (GBF); Graph signal processing; Kernel-based interpolation; Positive definite functions; Quadrature on graphs; Space-frequency analysis on graphs; Spectral graph theory; Graph basis functions (GBF); Space-frequency analysis on graphs; Approximation errors for interpolation; Graph signal processing; Positive definite functions; Spectral graph theory; Quadrature on graphs; Kernel-based interpolation
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2022.03.005

For the interpolation of graph signals with generalized shifts of a graph basis function (GBF), we introduce the concept of positive definite functions on graphs. This concept merges kernel-based interpolation with spectral theory on graphs and can be regarded as a graph analog of radial basis function interpolation in Euclidean spaces or spherical basis functions. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner-type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF's and to study GBF interpolation in more detail. We are able to characterize the native spaces of the interpolants. We provide explicit estimates for the interpolation error and obtain bounds for the numerical stability. As a final application, we show how GBF interpolation can be used to get quadrature formulas on graphs.