Harmonic Extension

In this paper, we consider the harmonic extension problem, w hich is widely used in many applications of machine learning. We find that th e ransitional method of graph Laplacian fails to produce a good approximation of t he classical harmonic function. To tackle this problem, we propose a new method cal led the point integral method (PIM). We consider the harmonic extension probl em 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, wh ich is easy to be discretized from points. Based on the integral equation, we exp lain the reason why the transitional graph Laplacian may fail to approximate the ha rmonicity in the classical sense and propose a different approach which we call th e volume constraint method (VCM). Theoretically, both the PIM and the VCM comput es a harmonic function with convergence guarantees, and practically, th ey are both simple, which amount to solve a linear system. One important application o f the harmonic extension in machine learning is semi-supervised learning. We ru n a popular semisupervised learning algorithm by Zhu et al. [16] over a coupl e of well-known datasets and compare the performance of the aforementioned approaches. Our experiments show the PIM performs the best.