In this talk we present an algorithm for manifold learning
and nonlinear dimensionality reduction. Based on a set of
unorganized data points sampled with noise from a parameterized
manifold, the local geometry of the manifold is learned by
constructing an approximation for the tangent space at
each data point, and those tangent spaces are then
aligned to give the global coordinates of the data
points with respect to the underlying manifold. We also
present a spectral analysis of the alignment matrix and show
how it leads to domain-decomposition type of algorithms. As applications
of the proposed methods, we consider occlusion removal in
computer vision and energy landscape interpolation in molecular
dynamics simulations.
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