Harmonic Extension on The Point Cloud

In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We formulate the harmonic extension as solving a Laplace--Beltrami equation with Dirichlet boundary condition. We use the point integral method (PIM) proposed in [Z. Li, Z. Shi, and J. Sun, Commun. Comput. Phys., 22 (2017), pp. 228--258; Z. Shi and J. Sun, Res. Math. Sci., to appear; Z. Li and Z. Shi, Multiscale Model. Simul., 14 (2016), pp. 874--905] to solve the Laplace--Beltrami equation. The basic idea of the PIM method is to approximate the Laplace equation using an integral equation, which is easy to discretize from points. Based on the integral equation, we found that the traditional graph Laplacian method (GLM) fails to approximate the harmonic functions near the boundary. One important application of the harmonic extension in machine learning is semisupervised learning. We run a popular semisupervised learning algorithm by Zhu, Ghahramani, and Lafferty [Machine Learning, Proce...