1. Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT

2. Manifold learning and manifold hypothesis [Kambhatla-Leen’93, Tannenbaum et al’00, Roweis-Saul’00, Belkin-Niyogi’03, Donoho-Grimes’04]

3. When is the Manifold Hypothesis true?

4. Reach of a submanifold of R n Large reach Small reach reach

5. Low dimensional manifolds with bounded volume and reach

6. Testing the Manifold Hypothesis

7. Sample Complexity of testing the manifold hypothesis [

8. Algorithmic question

9. Sample complexity of testing the Manifold Hypothesis

10. Empirical Risk Minimization

11. Fitting manifolds TexPoint Display

12. Reduction to k-means

13. Proving a Uniform bound for k-means

14. Fat-shattering dimension

15. Bound on sample complexity

16. VC dimension

18. Random projection

20. Bound on sample complexity

21. Fitting manifolds

22. Algorithmic question

24. Outline

27. (3) Generating a smooth vector bundle

29. Outline

30. (4) Generating a putative manifold

34. Outline

35. (5) Bundle map

37. Outline

39. Concluding Remarks An algorithm for testing the manifold hypothesis. Future directions: (a)Make practical and test on real data (b)Improve precision in the reach – get rid of controlled constants depending on d. (c)Better algorithms under distributional assumptions

40. Thank You!