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Manifold learning

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

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  1. Manifold learning Xin Yang Data Mining Course

  2. Outline • Manifold and Manifold Learning • Classical Dimensionality Reduction • Semi-Supervised Nonlinear Dimensionality Reduction • Experiment Results • Conclusions Data Mining Course

  3. What is a manifold? Data Mining Course

  4. Examples: sphere and torus Data Mining Course

  5. Why we need manifold? Data Mining Course

  6. Data Mining Course

  7. 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. Data Mining Course

  8. Manifold learning • Intrinsic Dimensionality Estimation • Dimensionality Reduction Data Mining Course

  9. 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 Data Mining Course

  10. Semi-supervised NDR • Prior information Can be obtained from experts or by performing experiments Eg: moving object tracking Data Mining Course

  11. Semi-supervised NDR • Assumption: Assuming the prior information has a physical meaning, then the global low dimensional coordinates bear the same physical meaning. Data Mining Course

  12. Basic LLE Data Mining Course

  13. Basic LTSA • Characterized the geometry by computing an approximate tangent space Data Mining Course

  14. SS-LLE & SS-LTSA • Give m the exact mapping data points . • Partition Y as • Our problem : Data Mining Course

  15. SS-LLE & SS-LTSA • To solve this minimization problem, partition M as: • Then the minimization problem can be written as Data Mining Course

  16. SS-LLE & SS-LTSA • Or equivalently • Solve it by setting its gradient to be zero, we get: Data Mining Course

  17. 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 Data Mining Course

  18. Sensitivity Analysis • The sensitivity of the solution depends on the condition number of the matrix Data Mining Course

  19. Inexact Prior Information • Add a regularization term, weighted with a parameter Data Mining Course

  20. Inexact Prior Information • Its minimizer can be computed by solving the following linear system: Data Mining Course

  21. 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 Data Mining Course

  22. Incomplete Tire Data Mining Course

  23. Data Mining Course

  24. Relative error with different number of prior points Data Mining Course

  25. Up body tracking Data Mining Course

  26. Results of SSLTSA Data Mining Course

  27. Results of inexact prior information algorithm Data Mining Course

  28. Conclusions • Manifold and manifold learning • Semi-supervised manifold learning • Future work Data Mining Course

  29. Thank you ! Data Mining Course

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