  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.
