  We consider the problem of learning a manifold from a teacher's
demonstration. Extending existing approaches of learning from randomly sampled
data points, we consider contexts where data may be chosen by a teacher. We
analyze learning from teachers who can provide structured data such as
individual examples (isolated data points) and demonstrations (sequences of
points). Our analysis shows that for the purpose of teaching the topology of a
manifold, demonstrations can yield remarkable decreases in the amount of data
points required in comparison to teaching with randomly sampled points. We also
discuss the implications of our analysis for learning in humans and machines.
