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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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Thank you !

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