1. EE 290A: Generalized Principal Component Analysis Lecture 4: Generalized Principal Component Analysis Sastry & Yang © Spring, 2011EE 290A, University of California, Berkeley1

2. This lecture GPCA: Problem Definition Segmentation of Multiple Hyperplanes Reminder: HW 1 due on Feb. 8 th. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 2

3. Problem Definition Define a mixture subspace model Subspace Segmentation Problem: Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 3

4. Projectivization of Affine Subspaces Every affine subspace can be “lifted” to a linear subspace by adding the homogeneous coordinates Homogeneous representation Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 4

5. Conclusion: Projectivization does not lose information on data model and sample membership Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 5

6. Subspace Projection High-dim data may lie in low-dim subspaces When d << D, estimation is not efficient Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 6 Images of a subject under illumination lie on a 20-dim subspace

7. Subspace-Preserving Projections Subspaces in high-D space can be projected onto a lower-D space while the membership of the samples is preserved Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 7

8. If the span of all subspaces is still a proper subspace of the ambient space : use PCA If the span is the whole space, yet the largest dimension is less than (D-1) Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 8

9. The approach for mixture-subspace segmentation Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 9

10. Choosing a SP-Projection Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 10

11. 3.2 Introductory Cases Segmenting points on a line Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 11

12. Determine the number of groups Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 12 Question: When j=K, is the null space of P always 1-D in this case?

13. Segmenting lines on a plane Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 13

14. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 14

15. Question 1: How to determine the number of lines? Question 2: When k=K, is the null space of V always rank-1? Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 15

16. Segmenting point clusters on a line or segmenting lines on a plane is a special case of mixture hyperplanes. Segmenting multiple hyperplanes Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 16

17. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 17

18. Find the vanishing polynomial from embedded data Determine the number of hyperplanes by the rank of the embedded data matrix V. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 18

19. Recover subspaces from vanishing polynomial Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 19

20. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 20

21. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 21