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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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### Manifold learning

Xin Yang

Data Mining Course

• Manifold and Manifold Learning

• Classical Dimensionality Reduction

• Semi-Supervised Nonlinear Dimensionality Reduction

• Experiment Results

• Conclusions

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• 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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• Intrinsic Dimensionality Estimation

• Dimensionality Reduction

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• 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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• Prior information

Can be obtained from experts or by performing experiments

Eg: moving object tracking

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• Assumption:

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

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• Characterized the geometry by computing an approximate tangent space

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• Give m the exact mapping data points .

• Partition Y as

• Our problem :

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• To solve this minimization problem, partition M as:

• Then the minimization problem can be written as

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• Or equivalently

• Solve it by setting its gradient to be zero, we get:

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

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• Add a regularization term, weighted with a parameter

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

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• “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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• Manifold and manifold learning

• Semi-supervised manifold learning

• Future work

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