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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) 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 be discretized from points. Based on the integral equation, we found that traditional graph Laplacian method (GLM) may fail to approximate the harmonic functions in the classical sense. For the Laplace-Beltrami equation with Dirichlet boundary, we can prove the convergence of the point integral method. The point integral method is also very easy to implement, which only requires a minor modification of the graph Laplacian. 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. We also apply PIM to an image recovery problem and show it outperforms GLM. Finally, on a model problem of Laplace-Beltrami equation with Dirichlet boundary, we prove the convergence of the point integral method.

Harmonic extension; Point cloud; Point integral method; Graph Laplacian