1. Principal Component Analysis (PCA)
Dated back to Pearson (1901)
A set of data are summarized as a linear combination of an ortonormal set of vectors which define a new coordinate system.
Figure from Hastie, et, al 2001

2. Principal Components Analysis (PCA)

3. Principal Components Analysis (PCA)

4. Singular Value Decomposition

5. Singular Value Decomposition

6. Singular Value Decomposition

7. Singular Value Decomposition

8. Example 1
Use the data set "noisy.mat" available on the course web page. The data set consists of 1965, 20-pixel-by-28-pixel grey-scale images distorted by adding Gaussian noises to each pixel with s=25.

9. Example 1
Apply PCA to the noisy data. Suppose the intrinsic dimensionality of the data is 10. Compute reconstructed images using the top d = 10 eigenvectors and plot original and reconstructed images

10. Example 1
If original images are stored in matrix X (it is 560 by 1965 matrix) and reconstructed images are in matrix X_hat , you can type in
colormap gray
and then
imagesc(reshape(X(:, 10), 20 28)’)
imagesc(reshape(X_hat(:, 10), 20 28)’)
to plot the 10th original image and its reconstruction.

11. Example 2

12. Example 2 Load the sample data, which includes digits 2 and 3 of
64 measurements on a sample of 400.
load 2_3.mat
Extract appropriate features by PCA
[u s v]=svd(X','econ');
Create data
Low_dimensional_data=u(:,1:2);
Observe low dimensional data
Imagesc(Low_dimensional_data)

13. Kernel Methods

14. Kernel Trick

15. Observed and Feature