Diffusion based Gaussian process regression via heat kernel reconstruction
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We propose an algorithm for Gaussian Process regression on an unknown embedded manifold in a Euclidean space by using the heat kernel of the manifold as the covariance kernel in the Gaussian Process. It is not straightforward to define a valid covariance kernel for the Gaussian process regression when the predictors are on a manifold. Although the heat kernel provides a natural and canonical choice theoretically, it is analytically intractable to directly evaluate, and current Bayesian methods rely on computationally demanding Monte Carlo approximations. We instead develop a computationally efficient and accurate approximation to the heat kernel relying on the diffusion property of the graph Laplacian, leading to a diffusion-based Gaussian process (DB-GP) modeling framework.In the manifold setting, we provide a series of spectral convergence results quantifying how the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator in the L^∞ sense.The convergence rate is also provided. Based on these results, convergence of the proposed heat kernel approximation algorithm, as well as the convergence rate, to the exact heat kernel is guaranteed.To our knowledge, this is the first work providing a numerical heat kernel reconstruction from the point cloud with theoretical guarantees. We also discuss the performance when there are measurement errors in the predictors, so that they do not fall exactly on the embedded manifold. Simulations illustrate performance gains for the proposed approach over traditional approaches.
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