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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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