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Robust Fisher Discriminant Analysis

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Robust Fisher Discriminant Analysis Article presented at NIPS 2005 By Seung-Jean Kim, Alessandro Magnani, Stephen P. Boyd Presenter: Erick Delage – PowerPoint PPT presentation

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Title: Robust Fisher Discriminant Analysis


1
Robust Fisher Discriminant Analysis
  • Article presented at NIPS 2005
  • By Seung-Jean Kim, Alessandro Magnani, Stephen P.
    Boyd
  • Presenter Erick Delage
  • February 14, 2006

2
Outline
  • Background on Fisher Linear Discriminant Analysis
  • Making the approach robust to small sample sets
    while maintaining computation efficiency
  • Experimental results

3
Fisher Discriminant Analysis
  • Given two Random Variables X,Y ??n, find the
    linear discriminant ? ??n that maximizes Fishers
    discriminant ratio
  • Unique solution
  • Easy to compute
  • Probabilistic interpretation
  • Kernelizable
  • Naturally extends to k-class problems

4
Probabilistic Interpretation
5
Using Kernels
  • When discriminating in feature space ?(x).
  • We can use kernels
  • And show that ? is of the form
  • A projection along ? given by
  • And find ? by solving

6
Robust Fisher Discriminant Analysis
  • Uncertainty in (?x, ?x) (?y, ?y)
  • FDA is sensitive to estimation errors of these
    parameters.
  • Can we make it more robust using general convex
    uncertainty models on the problem data?
  • Is it still a computationally feasible technique?

7
Max Worst-case Fisher Discriminant ratio
8
Max Worst-case Fisher Discriminant ratio
9
Max Worst-case Fisher Discriminant ratio
Given ,
10
Experimental Results
  • Two benchmark problems from the machine learning
    repository
  • Sonar 208 points, n 60
  • Ionosphere 351 points, n 34
  • Uncertainty models

11
Experimental Results
12
References
  • S.-J. Kim, A. Magnani, and S. P. Boyd Robust
    Fisher Discrminant Analysis. In T. Leen, T.
    Dietterich and V. Tresp, editors, Advances in
    Neural Information Processing Systems, 18, pp
    659-666 ,MIT Press, 2006.
  • S. Mika, G. Rätsch, and K.-R. Müller A
    Mathematical Programming Approach to the Kernel
    Fisher Algorithm. In T. Leen, T. Dietterich and
    V. Tresp, editors, Advances in Neural Information
    Processing Systems, pp 591-597,MIT Press, 2000.
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