# Principal Component Analysis

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```					Principal Component Analysis

IML 2004-5
Outline

• Max the variance of the output coordinates

• Optimal reconstruction

• Generating data

• Limitations of PCA
Variance in Face Pictures
Eigenfaces

•Figure/ground
•Orientation
•Lighting
•Hairline
Eigenfaces
100 images
30x30 pixels
subtract
mean

900   A
100

900     AAT
900
Maximizing Output Variance

The first eigenvector
(highest eigenvalue)
characterizes the maximal
variance in the image:
figure - background
Maximizing Output Variance

The second eigenvector
characterizes right orientation
Maximizing Output Variance
Maximizing Output Variance

Variance

Dimensionality
Optimal Reconstruction

q=1       q=2    q=4    q=8

Original
q=16      q=32   q=64   q=100…    Image
If n>>m
e.g. n=80x80 pixels >> m=100 images
Problem: finding the eigenvectors of a
6400x6400 matrix = O(64003)
Solution: extract the eigenvectors Q of ATA
If n>m

q=1    q=2    q=4    q=8

Original
q=16   q=32   q=64   q=100    Image
Generating Data
Generating Data
Limits of PCA
Should the goal be finding independent rather
than pair-wise uncorrelated dimensions
•Independent Component Analysis (ICA)

ICA       PCA
Limits of PCA
Are the maximal variance dimensions the
relevant dimensions for preservation?
•Relevant Component Analysis (RCA)
•Fisher Discriminant analysis (FDA)

```
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