Testing the Manifold Hypothesis

Hari NarayananUniversity 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

Large reach Small reach

M reach

Low dimensional manifolds with bounded volume and reach

Testing the Manifold Hypothesis

Sample Complexity of testing the manifold hypothesis

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Algorithmic question

Sample complexity of testing the Manifold Hypothesis

Empirical Risk Minimization

Fitting manifolds

TexPoint 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!