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 [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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Random graphs are mathematical models that have applications in a wide range of domains. Among them, the Erdős-Rényi and the random geometric graphs are two models whose theoretical properties (e.g., clique number and the largest component) are perhaps most well studied. We are interested in mixed models coming from the overlay of two graph structures. In particular, we study the following model where one adds Erdős-Rényi (ER) type perturbation to a random geometric graph. More precisely, assume G * X is a random geometric graph sampled from a nice measure on a metric space X = (X, d). The input observed graph G(p, q) is generated by removing each existing edge from G * X with probability p, while inserting each non-existent edge to G * X with probability q. We refer to such random p-deletion and q-insertion as ER-perturbation. Although these graphs are related to the objects in the continuum percolation theory, our understanding of them is still rather limited.In this paper we consider a localized version of the classical notion of clique number for the aforementioned ER-perturbed random geometric graphs: Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. The clique number of the graph is simply the largest edge clique number. Interestingly, given a ER-perturbed random geometric graph, we show that the edge clique number presents two fundamentally different types of behaviors, depending on which "type" of randomness it is generated from.As an application of the above results, we show that by using a filtering process based on the edge clique number, we can recover the shortest-path metric of the random geometric graph G * X within a multiplicative factor of 3, from an ER-perturbed observed graph G(p, q), for a significantly wider range of insertion probability q than in previous work.

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Let G n be a random geometric graph, and then for q, p ∈ [0, 1) we construct a (q, p)-perturbed noisy random geometric graph G q,p n where each existing edge in G n is removed with probability q, while and each non-existent edge in G n is inserted with probability p. We give asymptotically tight bounds on the clique number 𝜔 ( G q,p n) for several regimes of parameter.