Interpolating splines on graphs for data science applications

TL;DR

This paper develops intrinsic interpolatory bases for graph-structured data, demonstrating their exponential decay properties and potential for kernel-based machine learning applications.

Contribution

It introduces polyharmonic Lagrange functions on graphs and analyzes their decay, providing new tools for data analysis on graph structures.

Findings

Lagrange functions exhibit exponential decay away from centers

Decay rate depends on zero density of the Lagrange functions

Lagrange bases are effective for kernel-based machine learning on graphs

Abstract

We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications.