Maximum likelihood estimation of intrinsic dimension
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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


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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