TITLE: Reference request for manifold learning
QUESTION [10 upvotes]: 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.

REPLY [5 votes]: This is not exactly what you're looking for, but on the subject of Topology in Computer Science, here are two recommendations I can make:

Topology and its Applications, William F. Basener, Wiley-Interscience, 2006
Topology for Computing, Afra Zomorodian, Cambridge University Press, 2006 

They both give some inkling into the Differential Topology aspects of Machine Learning. Also, not sure if you've already seen this, but here are some lectures from the likes of Smale on Machine Learning.