Harmonic Extension

TL;DR

This paper introduces the point integral method (PIM) for harmonic extension problems in machine learning, addressing limitations of graph Laplacian methods by solving PDEs on manifolds with convergence guarantees.

Contribution

The paper proposes the PIM and volume constraint method (VCM) as new approaches for harmonic extension, improving approximation accuracy and providing theoretical convergence guarantees.

Findings

PIM outperforms traditional graph Laplacian methods in experiments.

Both PIM and VCM are simple to implement, involving solving linear systems.

The methods are theoretically proven to converge to harmonic functions.

Abstract

In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We find that the transitional method of graph Laplacian fails to produce a good approximation of the classical harmonic function. To tackle this problem, we propose a new method called the point integral method (PIM). We consider the harmonic extension problem from the point of view of solving PDEs on manifolds. The basic idea of the PIM method is to approximate the harmonicity using an integral equation, which is easy to be discretized from points. Based on the integral equation, we explain the reason why the transitional graph Laplacian may fail to approximate the harmonicity in the classical sense and propose a different approach which we call the volume constraint method (VCM). Theoretically, both the PIM and the VCM computes a harmonic function with convergence guarantees, and practically, they are both simple, which amount to solve a linear system. One important application of the harmonic extension in machine learning is semi-supervised learning. We run a popular semi-supervised learning algorithm by Zhu et al. over a couple of well-known datasets and compare the performance of the aforementioned approaches. Our experiments show the PIM performs the best.