A very fertile approach to "learning" a manifold from point samples is to convert the problem into one of embedding a graph under metric constraints. Fidelity to the original manifold is an issue because the
(invented) graph and the (estimated) metric constraints are both sources of bias. I'll show how this can be addressed algebraically, yielding methods that can recover exact isometries for a useful class of developable manifolds. These methods generalize very naturally to new points, making them viable as a nonlinear alternative to linear subspace methods in signal processing.
Graph embeddings, however, can be very misleading if the data submanifold is not isomorphic to a low-dimensional ball. How would we know? I'll present a topology-revealing graph-partitioning problem, and show some preliminary computational results.
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