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Maximum likelihood estimation of intrinsic dimension. Authors: Elizaveta Levina & Peter J. Bickel presented by: Ligen Wang. Plan. Problem Some popular methods MLE approach Statistical behaviors Evaluation Conclusions. Problem. Facts:

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maximum likelihood estimation of intrinsic dimension

Maximum likelihood estimation of intrinsic dimension

Authors: Elizaveta Levina & Peter J. Bickel

presented by: Ligen Wang

slide2
Plan
  • Problem
  • Some popular methods
  • MLE approach
  • Statistical behaviors
  • Evaluation
  • Conclusions
problem
Problem
  • Facts:
    • Many real-life high-D data are not truly high-dimensional
    • Can be effectively summarized in a space of much lower dimension
  • Why discover this low-D structure?
    • Help to improve performance in classification and other applications
  • Our target:
    • How much is this lower dimension exactly, i.e., the intrinsic dimension
  • Importance of this lower dimension:
    • If our estimation is too low, features are collapsed onto the same dimension
    • If too high, the projection becomes noisy and unstable
some popular methods
Some popular methods
  • PCA
    • Decides the dimension by users by how much covariance they want to preserve
  • LLE
    • User provides the manifold dimension
  • ISOMAP
    • Provides error curves that can be ‘eyeballed’ to estimate dimension
  • Etc.
conclusions
Conclusions
  • MLE produces good results on a range of simulated (both non-noisy and noisy) and read datasets
  • Outperforms two other methods
  • Suffers from a negative bias for high dimensions
    • Reason: approximation is based on observations falling in a small sphere, which requires very large sample size when the dimension is high
    • Good news: in reality, the intrinsic dimensions are low for most interesting applications
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