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Maximum likelihood estimation of intrinsic dimensionPowerPoint Presentation

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:
- 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

- 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

- 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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