Out-of-sample extension of band-limited functions on homogeneous manifolds using diffusion maps

In this paper, we address the problem of function extension when the available data lies on a homogeneous manifold (i.e. the domain of the function is a homogeneous manifold embedded in the Euclidean space) and the function is band-limited. We solve this problem in the general case in which the manifold is unknown. We assume that we have sufficient labeled data to reconstruct the function from labeled data. We also assume that we have enough data (at least exponential in the intrinsic dimension of the manifold) to approximate the Laplace-Beltrami operator on the manifold. The proposed method has a closed form solution and consists of matrix multiplication and inversion. As the size of data approaches infinity, the proposed method converges to the optimal solution as long as the function values are known on an appropriate sampling set. Simulation results demonstrate the advantage of the proposed method over commonly used function extension methods.