Project 11: Determining the Intrinsic Dimensionality of a Distribution

1. Project 11: Determining the Intrinsic Dimensionality of a Distribution Okke Formsma, Nicolas Roussisand Per Løwenborg

2. Outline • About the project • What is intrinsic dimensionality? • How can we assess the ID? • PCA • Neural Network • Nearest Neighbour • Experimental Results

3. Why did we chose this Project? • We wanted to learn more about developing and experiment with algorithms for analyzing high-dimensional data • We want to see how we can implement this into a program

4. Papers N. Kambhatla, T. Leen, “Dimension Reduction by Local Principal Component Analysis” J. Bruske and G. Sommer, “Intrinsic Dimensionality Estimation with Optimally Topology Preserving Maps” P. Verveer, R. Duin, “An evaluation of intrinsic dimensionality estimators”

5. Howdoesdimensionalityreductioninfluenceour lives? • Compress images, audio and video • Redusing noise • Editing • Reconstruction

6. This is a image goingthroughdifferentsteps in a reconstruction

7. Intrinsic Dimensionality The number of ‘free’ parameters needed to generate a pattern Ex: • f(x) = -x² => 1 dimensional • f(x,y) = -x² => 1 dimensional

8. Principal Component analysis

9. Principal Component Analysis (PCA) • The classic technique for linear dimension reduction. • It is a vector space transformation which reduce multidimensional data sets to lower dimensions for analysis. • It is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences.

10. Advantages of PCA • Since patterns in data can be hard to find in data of high dimension, where the luxury of graphical representation is not available, PCA is a powerful tool for analysing data. • Once you have found these patterns in the data, you can compress the data, -by reducing the number of dimensions- without much loss of information.

11. Example

12. Problems with PCA • Data might be uncorrelated, but PCA relies on second-order statistics (correlation), so sometimes it fails to find the most compact description of the data.

13. Problems with PCA

14. First eigenvector

15. Second eigenvector

16. A better solution?

17. Local eigenvector

18. Local eigenvectors

19. Local eigenvectors

20. Another problem

21. Is this the principal eigenvector?

22. Or do we need more than one?

23. Choose

24. The answer depends on your application Low resolution High resolution

25. Challenges • How to partition the space? • How many partitions should we use? • How many dimensions should we retain?

26. How to partition the space? Vector Quantization Lloyd Algorithm Partition the space in k sets Repeat until convergence: Calculate the centroids of each set Associate each point with the nearest centroid

27. Lloyd Algorithm Step 1: randomly assign Set 1 Set 2

28. Lloyd Algorithm Step 2: Calculate centriods Set 1 Set 2

29. Lloyd Algorithm Step 3: Associate points with nearest centroid Set 1 Set 2

30. Lloyd Algorithm Step 2: Calculate centroids Set 1 Set 2

31. Lloyd Algorithm Step 3: Associate points with nearest centroid Set 1 Set 2

32. Lloyd Algorithm Result after 2 iterations: Set 1 Set 2

33. How many partitions should we use? Bruske & Sommer: “just try them all” For k = 1 to k ≤ dimension(set): Subdivide the space in k regions Perform PCA on each region Retain significant eigenvalues per region

34. Which eigenvalues are significant? Depends on: • Intrinsic dimensionality • Curvature of the surface • Noise

35. Which eigenvalues are significant? Discussed in class: • Largest-n In papers: • Cutoff after normalization (Bruske & Summer) • Statistical method (Verveer & Duin)

36. Which eigenvalues are significant? Cutoff after normalization µx is the xth eigenvalue With α = 5, 10 or 20.

37. Which eigenvalues are significant? Statistical method (Verveer & Duin) Calculate the error rate on the reconstructed data if the lowest eigenvalue is dropped Decide whether this error rate is significant

38. Results • One dimensional space, embedded in 256*256 = 65,536 dimensions • 180 images of rotatingcylinder • ID = 1

39. Results

40. Neural Network PCA

41. Basic Computational Element - Neuron • Inputs/Outputs, Synaptic Weights, Activation Function

43. 3-Layer Autoassociators • N input, N output and M<N hidden neurons. • Drawbacks for this model. The optimal solution remains the PCA projection.

44. 5-Layer Autoassociators • Neural Network Approximators for principal surfaces using 5-layers of neurons. • Global, non-linear dimension reduction technique. • Succesfull implementation of nonlinear PCA using these networks for image and speech dimension reduction and for obtaining concise representations of color.

45. Third layer carries the dimension reduced representation, has width M<N • Linear functions used for representation layer. • The networks are trained to minimize MSE training criteria. • Approximators of principal surfaces.

46. Locally Linear Approach to nonlinear dimension reduction (VQPCA Algorithm) • Much faster than to train five-layer autoassociators and provide superior solutions. • This algorithm attempts to minimize the MSE (like 5-layers autoassociators) between the original data and its reconstruction from a low-dimensional representation. (reconstruction error)

47. VQPCA • 2 Steps in Algorithm: • Partition the data space by VQ (clustering). • Performs local PCA about each cluster center. VQPCA is actually a local PCA to each cluster.

48. We can use 2 kinds of distances measures in VQPCA:1) Euclidean Distance2) Reconstruction Distance Example intended for a 1D local PCA:

49. 5-layer Autoassociators vs VQPCA • Difficulty to train 5-layer autoassociators. Faster training in VQPCA algorithm. (VQ can be accelerated using tree-structured or multistage VQ) • 5-layer autoassociators are prone to trapping in poor local optimal. • VQPCA slower for encoding new data but much faster for decoding.