I am interested in learning about manifold learning (no pun intended) and would like to know of some references that discuss the subject from a more geometric perspective. By manifold learning I mean the idea of studying high dimensional data using techniques from geometry. I'm interested in knowing how topics from differential geometry and topology such as Hodge theory and Morse theory can be used to study questions in manifold learning. I thought I would ask if people have any recommendations for papers or books that explain these topics more from a more geometric perspective.
Update: I expect that there is no mythical survey paper that explains all aspects of manifold learning to someone that knows about geometry and topology. Specifically, I would be interested in knowing of some survey papers which explain how tools from Riemannian geometry would be useful in manifold learning. Perhaps how such tools can be used for nonlinear dimensionality reduction.