Locally supported, quasi-interpolatory bases for the approximation of functions on graphs

TL;DR

This paper introduces locally supported, quasi-interpolatory basis functions on graphs, derived from energy minimization problems, with theoretical error bounds and practical numerical validation for graph approximation tasks.

Contribution

It proposes a novel class of basis functions on graphs with local support, derived from spline-like functions, and provides error estimates and inequalities for their approximation properties.

Findings

Error bounds between local and Lagrange basis functions are established.

Numerical experiments validate the theoretical error estimates.

A new matrix inequality improves existing bounds for positive definite matrices.

Abstract

Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present $\ell_\infty$ error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies a mild assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate $\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$.