Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with...

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

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

Testing the Manifold Hypothesis

Hari NarayananUniversity of Washington

In collaboration with Charles Fefferman and Sanjoy Mitter

Princeton MIT

Page 2: 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]

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When is the Manifold Hypothesis true?

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Reach of a submanifold of Rn

Large reach Small reach

M reach

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Low dimensional manifolds with bounded volume and reach

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Testing the Manifold Hypothesis

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Sample Complexity of testing the manifold hypothesis

[

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Algorithmic question

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Sample complexity of testing the Manifold Hypothesis

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Empirical Risk Minimization

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Fitting manifolds

TexPoint Display

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Reduction to k-means

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Proving a Uniform bound for k-means

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Fat-shattering dimension

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Bound on sample complexity

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VC dimension

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VC dimension

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Random projection

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Random projection

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Bound on sample complexity

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Fitting manifolds

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Algorithmic question

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Page 24: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

Outline

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Outline

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Outline

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(3) Generating a smooth vector bundle

Page 28: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(3) Generating a smooth vector bundle

Page 29: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

Outline

Page 30: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(4) Generating a putative manifold

Page 31: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(4) Generating a putative manifold

Page 32: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(4) Generating a putative manifold

Page 33: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(4) Generating a putative manifold

Page 34: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

Outline

Page 35: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(5) Bundle map

Page 36: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

(5) Bundle map

Page 37: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

Outline

Page 38: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

Outline

Page 39: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

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

Page 40: Testing the Manifold Hypothesis Hari Narayanan University of Washington In collaboration with Charles Fefferman and Sanjoy Mitter Princeton MIT.

Thank You!


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