Learning a manifold from a teacher's demonstrations

We consider the problem of learning a manifold. Existing approaches learn by approximating the manifold directly or the topology, but require large amounts of data to overcome challenges posed by manifolds with small reach and non-uniform sampling. We consider contexts where some data could be marked by a teacher. We consider the problem of learning from a perfectly knowledgeable teacher, providing bounds on the sample complexity for learning the manifold exactly and contrast with learning only up to topology. We then consider learning from a teacher with partial knowledge, in which a Topological Data Analysis learner's inference integrates observations with demonstrations provided by the teacher. Examples on simulated and real data illustrate how teaching can facilitate learning the topology and geometry of the manifold.