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Abstract:
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In analyzing complex datasets, it is often of interest to infer lower dimensional structure underlying the higher dimensional observations. As a flexible class of nonlinear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. In this talk, we propose a new methodology for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing application of Gaussian processes for probabilistic manifold reconstruction. We also provide simulated and real data examples to illustrate the performance. This talk is based on the joint work with David Dunson.
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