Manifold%20learning

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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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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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Thank you !