Manifold learning

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# Manifold learning - PowerPoint PPT Presentation

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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Presentation Transcript

### Manifold learning

Xin Yang

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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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Manifold learning
• Intrinsic Dimensionality Estimation
• Dimensionality Reduction

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

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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
• Add a regularization term, weighted with a parameter

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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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Up body tracking

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Results of SSLTSA

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

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Thank you !

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