Graph signal interpolation with Positive Definite Graph Basis Functions

TL;DR

This paper introduces positive definite graph basis functions for graph signal interpolation, merging kernel methods with spectral graph theory, enabling stable interpolation, error estimation, and quadrature on graphs.

Contribution

It develops a new framework of positive definite functions on graphs, extending radial basis function concepts to graph domains, with theoretical analysis and practical applications.

Findings

Characterized native spaces of GBF interpolants

Provided explicit error estimates for interpolation

Demonstrated quadrature formulas on graphs

Abstract

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.