1. EE 290A: Generalized Principal Component Analysis Lecture 2 (by Allen Y. Yang): Extensions of PCA Sastry & Yang © Spring, 2011EE 290A, University of California, Berkeley1

2. Last time Challenges in modern data clustering problems. PCA reduces dimensionality of the data while retaining as much data variation as possible. Statistical view: The first d PCs are given by the d leading eigenvectors of the covariance. Geometric view: Fitting a d-dim subspace model via SVD Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 2

3. This lecture Determine an optimal number of PCs: d Probabilistic PCA Kernel PCA Robust PCA shall be discussed later Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 3

4. Determine the number of PCs Choosing the optimal number of PCs in noise-free case is straightforward: Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 4

5. In the noisy case Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 5 knee point

6. A Model Selection Problem With moderate Gaussian noise, to keep 100% fidelity of the data, all D-dim must be preserved. However, we can still find tradeoff between model complexity and data fidelity? Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 6

7. More principled conditions Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 7

8. Probabilistic PCA: A generative approach Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 8

9. Given sample statistics, (*) contains ambiguities Assume y is standard normal, and εis isotropic Then each observation is also Gaussian Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 9

10. Determining principal axes by MLE Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 10 Compute the log-likelihood for n samples The gradient of L leads to stationary points

11. Two nontrivial solutions Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 11

12. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 12

13. Kernel PCA: for nonlinear data Nonlinear embedding Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 13

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

15. Question: How to recover the coef? Compute the null space of the data matrix The special polynomial embedding is called the Veronese map Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 15

16. Dimensionality Issue in Embedding Given D and order n, what is the dimension of the Veronese map? Often the dimension blows up with large D or n. Question: Can we find the higher-order nonlinear structures without explicitly calling the embedding function? Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 16

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

18. In the case M is much larger than n Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 18

19. Kernel PCA Computations in NLPCA only involve inner products of the embedded samples, not the samples themselves. Therefore, the mapping relation can be expressed in the the computation of PCA without explicitly calling the embedding function. The inner product of two embedded samples is called the kernel function. Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 19

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

21. Computing NLPCs via Kernel Matrix Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 21

22. Examples of Popular Kernels Polynomial kernel: Gaussian kernel (Radial Basis Function): Intersection kernel: Sastry & Yang © Spring, 2011 EE 290A, University of California, Berkeley 22