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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.

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Sorry I should be more clear about manifold learning. I mean the idea of studying high dimensional data using techniques from geometry. –  Chirag Lakhani Mar 9 '11 at 18:19
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Have you looked at the surveys by Carlsson or Harer and Edelsbrunner? There are a lot of resources at comptop.stanford.edu –  Sam Isaacson Mar 9 '11 at 18:48
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The Wikipedia article (en.wikipedia.org/wiki/Manifold_learning) contains 25 references and 9 external links. Perhaps you could use this resource to sharpen your question? –  Joseph O'Rourke Mar 9 '11 at 20:16
    
Thanks for your comments I will try to be more precise. –  Chirag Lakhani Mar 9 '11 at 22:31
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Partha Niyogi wrote a number of papers about using the graph laplacian of a data set for learning purposes. In the limit of infinite data living on a manifold, this converges to the Laplace-Beltrami operator. –  Simon Lyons Mar 10 '11 at 12:09
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I came across a nice video lecture by Niyogi that gives a nice survey of manifold learning. I thought I would share in case anyone else was interested.

http://videolectures.net/mlss09us_niyogi_belkin_gmml/

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Here's the web page for a seminar on this stuff we ran at Wisconsin, featuring a list of references at the top. I think Gunnar Carlsson's expose is very well-written and interesting, though it's certainly more about algebraic topology than differential geometry (i.e. the goal is to compute homology, not differential invariants like curvature.) The work of Smale, Niyogi, and Weinberger (for instance, this paper) approaches the same problem from a slightly different point of view and is also really interesting.

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Thanks for that website and for the Smale, Niyogi, and Weinberger reference it looks very interesting. –  Chirag Lakhani Mar 25 '11 at 14:29
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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:

  1. Topology and its Applications, William F. Basener, Wiley-Interscience, 2006
  2. 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.

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Thanks! I didn't know about the Topology and its Applications books. Looks interesting. –  Chirag Lakhani Apr 10 '11 at 15:14
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There are of course many interesting and good textbooks on differential geometry around. Personally, I like very much the one of Michor (published in the AMS, but there are chances that he has some pdf on his homepage). However, this is a rather generic differential geometry textbook. Also the books of Lang and of Lee (both GTM Springer) are usefull and cover a lot of material.

For Hodge theory on compact Riemannian manifolds, I think the still valid standard textbook is Warner's book (in GTM Springer), though this books is in some aspects (notation...) not quite what I personally like.

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I think the OP wants a reference which is directly related to manifold learning, not a textbook on differential geometry. –  J.C. Ottem Mar 9 '11 at 18:39
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Oops. I really misunderstood that. Sorry, you should ignore the answer (though Michor's book is still very good) –  Stefan Waldmann Mar 10 '11 at 7:47
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