1. Maximum likelihood estimation of intrinsic dimension
Authors: Elizaveta Levina & Peter J. Bickel
presented by: Ligen Wang

2. Plan Problem Some popular methods MLE approach Statistical behaviors
Evaluation
Conclusions

3. Problem Facts: Why discover this low-D structure? Our target:
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

4. Some popular methods PCA LLE ISOMAP Etc.
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.

5. MLE approach

6. A little math

7. Statistical behaviors

8. Evaluation – 1: on manifold

9. Evaluation – 2: near manifold

10. Evaluation – 3: real-world data

11. 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