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Manifold learning. Xin Yang. Outline. Manifold and Manifold Learning Classical Dimensionality Reduction Semi-Supervised Nonlinear Dimensionality Reduction Experiment Results Conclusions. What is a manifold?. Examples: sphere and torus. Why we need manifold?. Manifold learning.

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manifold learning

Manifold learning

Xin Yang

Data Mining Course

outline
Outline
  • Manifold and Manifold Learning
  • Classical Dimensionality Reduction
  • Semi-Supervised Nonlinear Dimensionality Reduction
  • Experiment Results
  • Conclusions

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what is a manifold
What is a manifold?

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examples sphere and torus
Examples: sphere and torus

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why we need manifold
Why we need manifold?

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manifold learning1
Manifold learning
  • Raw format of natural data is often high dimensional, but in many cases it is the outcome of some process involving only few degrees of freedom.

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manifold learning2
Manifold learning
  • Intrinsic Dimensionality Estimation
  • Dimensionality Reduction

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dimensionality reduction
Dimensionality Reduction
  • Classical Method:

Linear: MDS & PCA (Hastie 2001)

Nonlinear: LLE (Roweis & Saul, 2000) ,

ISOMAP (Tenebaum 2000),

LTSA (Zhang & Zha 2004)

-- in general, low dimensional coordinates lack physical meaning

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semi supervised ndr
Semi-supervised NDR
  • Prior information

Can be obtained from experts or by performing experiments

Eg: moving object tracking

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semi supervised ndr1
Semi-supervised NDR
  • Assumption:

Assuming the prior information has a physical meaning, then the global low dimensional coordinates bear the same physical meaning.

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basic lle
Basic LLE

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basic ltsa
Basic LTSA
  • Characterized the geometry by computing an approximate tangent space

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ss lle ss ltsa
SS-LLE & SS-LTSA
  • Give m the exact mapping data points .
  • Partition Y as
  • Our problem :

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ss lle ss ltsa1
SS-LLE & SS-LTSA
  • To solve this minimization problem, partition M as:
  • Then the minimization problem can be written as

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ss lle ss ltsa2
SS-LLE & SS-LTSA
  • Or equivalently
  • Solve it by setting its gradient to be zero, we get:

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sensitivity analysis
Sensitivity Analysis
  • With the increase of prior points, the condition number of the coefficient matrix gets smaller and smaller, the computed solution gets less sensitive to the noise in and

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sensitivity analysis1
Sensitivity Analysis
  • The sensitivity of the solution depends on the condition number of the matrix

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inexact prior information
Inexact Prior Information
  • Add a regularization term, weighted with a parameter

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inexact prior information1
Inexact Prior Information
  • Its minimizer can be computed by solving the following linear system:

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experiment results
Experiment Results
  • “incomplete tire”

--compare with basic LLE and LTSA

--test on different number of prior points

  • Up body tracking

--use SSLTSA

--test on inexact prior information algorithm

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incomplete tire
Incomplete Tire

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up body tracking
Up body tracking

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results of ssltsa
Results of SSLTSA

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conclusions
Conclusions
  • Manifold and manifold learning
  • Semi-supervised manifold learning
  • Future work

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slide29

Thank you !

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