Testing the Manifold Hypothesis Hari Narayanan University of

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

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

When is the Manifold Hypothesis true?

Reach of a submanifold of Rn reach Large reach Small reach

Low dimensional manifolds with bounded volume and reach

Testing the Manifold Hypothesis

Sample Complexity of testing the manifold hypothesis

Algorithmic question

Sample complexity of testing the Manifold Hypothesis

Empirical Risk Minimization

Fitting manifolds Tex. Point Display

Reduction to k-means

Proving a Uniform bound for k-means

Fat-shattering dimension

Bound on sample complexity

VC dimension

VC dimension

Random projection

Random projection

Bound on sample complexity

Fitting manifolds

Algorithmic question

Outline

Outline

Outline

(3) Generating a smooth vector bundle

(3) Generating a smooth vector bundle

Outline

(4) Generating a putative manifold

(4) Generating a putative manifold

(4) Generating a putative manifold

(4) Generating a putative manifold

Outline

(5) Bundle map

(5) Bundle map

Outline

Outline

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

Thank You!