Subspace Clustering

1. Subspace Clustering Ali Sekmen Computer Science College of Engineering Tennessee State University 1st Annual Workshop on Data Sciences

2. Outline • Subspace Segmentation Problem • Motion Segmentation • Principal Component Analysis • Dimensionality Reduction • Spectral Clustering • Presenter • Dr. Ali Sekmen

3. Subspace Segmentation • In many engineering and mathematics applications, data lives in a union of low dimensional subspaces • Motion segmentation • Facial images of a person with the same expression under different illumination approximately lie on the same subspace

4. Face Recognition

5. Problem Statement

6. Problem Statement

7. Problem Statement

8. What are we trying to solve?

9. Example – Motion Segmentation

10. Motion Segmentation Motion segmentation problem can simply be defined as identifying independently moving rigid objects in a video.

11. Motion Segmentation We will show that all trajectories lie in a 4-dim subspace of

12. Motion Segmentation Z Z p z x Y Y X y X

13. Motion Segmentation Z p z x Y X y

14. Motion Segmentation Z p z x Y X y

15. Motion Segmentation

16. Motion Segmentation Y X

17. Motion Segmentation Motion Segmentation Y X

18. Motion Segmentation

19. Motion Segmentation

20. Principal Component Analysis • The goal is to reduce dimension of dataset with minimal loss of information • We project a feature space onto a smaller subspace that represent data well • Search for a subspace which maximizes the variance of projected points • This is equivalent to linear least square fitting • Minimize the sum of squared distances between points and subspace • We find directions (components) that maximizes variance in dataset • PCA can be done by • Eigenvalue decomposition of a data covariance matrix • Or SVD of a data matrix

21. Least Square Approximation

22. Principal Component Analysis

23. Principal Component Analysis

24. PCA with SVD Coordinates w.r.t. new basis

25. Principal Component Analysis inch cm

26. Principal Component Analysis

27. Solution with SVD

28. PCA: Pre-Processing

29. PCA: Optimization

30. PCA: Reduce Dimensionality

31. PCA: Reduce Dimensionality

32. General PCA

33. Spectral Clustering • A very powerful clustering algorithm • Easy to implement • Outperforms traditional clustering algorithms • Example: k-means • It is not easy to understand why it works • Given a set of data points and some similarity measure between all pairs of data points, we divide data into groups • Points in the same group are similar • Points in different groups are dissimilar

34. Spectral Clustering • Most of subspace clustering algorithms employ spectral clustering as the last step

35. Similarity

36. Spectral Clustering

37. Spectral Clustering

38. Spectral Clustering

39. Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg

40. Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg

41. Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg

42. Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg

43. Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg

44. Spectral Clustering Example

45. Spectral Clustering Example

46. Spectral Clustering Example