Harmonic Extension on Point Cloud ∗
نویسندگان
چکیده
Abstract. 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 LaplaceBeltrami equation with Dirichlet boundary condition. We use the point integral method (PIM) proposed in [14, 19, 13] 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) fails to approximate the harmonic functions near the boundary. One important application of the harmonic extension in machine learning is semi-supervised learning. We run a popular semisupervised learning algorithm by Zhu et al. [24] 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 the model problem of Laplace-Beltrami equation with Dirichlet boundary, we prove the convergence of the point integral method.
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