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## PowerPoint Slideshow about ' Manifold learning' - erma

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Outline

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

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What is a manifold?

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Examples: sphere and torus

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Why we need manifold?

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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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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

- Prior information

Can be obtained from experts or by performing experiments

Eg: moving object tracking

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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

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SS-LLE & SS-LTSA

- Give m the exact mapping data points .
- Partition Y as
- Our problem :

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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-LTSA

- Or equivalently
- Solve it by setting its gradient to be zero, we get:

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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 Analysis

- The sensitivity of the solution depends on the condition number of the matrix

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Inexact Prior Information

- Its minimizer can be computed by solving the following linear system:

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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

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Relative error with different number of prior points

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Up body tracking

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Results of SSLTSA

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Results of inexact prior information algorithm

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Conclusions

- Manifold and manifold learning
- Semi-supervised manifold learning
- Future work

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