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Assignment 1 Dimensionality Reduction Solution
Introduction
In this assignment, in addition to related theory/math questions, you’ll work on visualizing data and reducing its dimensionality.
You may not use any functions from machine learning library in your code, however you may use statistical functions. For example, if available you MAY NOT use functions like
pca
entropy
however you MAY use basic statistical functions like:
std mean cov eig
Grading
Although all assignments will be weighed equally in computing your homework grade, below is the grading rubric we will use for this assignment:
Part 1
(Theory)
30pts
Part 2
(PCA)
40pts
Part 3
(Eigenfaces)
20pts
Report
10pts
TOTAL
100pts
Table 1: Grading Rubric
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DataSets
Yale Faces Datasaet This dataset consists of 154 images (each of which is 243x320 pixels) taken from 14 people at 11 di erent viewing conditions (for our purposes, the rst person was removed from the o cial dataset so person ID=2 is the rst person).
The lename of each images encode class information:
subject< ID .< condition
Data obtained from: http://cvc.cs.yale.edu/cvc/projects/yalefaces/yalefaces.html
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Theory Questions
(15 points) Consider the following data:
2
5
14
3
2
12
0
3
6
2
7
6
5
Class 1 =
0
3
, Class 2 =
1
37
6
7
6
7
6
3
1
7
6
5
1
7
8
11
6
1
6
7
6
7
4
5
4
5
Compute the information gain for each feature. You could standardize the data overall, although it won’t make a di erence. (13pts).
Which feature is more discriminating based on results in Part (a) (2pt)?
(15 points) In principle component analysis (PCA) we are trying to maximize the variance of the data after projection while minimizing how far the magnitude of w, jwj is from being unit length. This results in attempting to nd the value of w that maximizes the equation
wT w (wT w 1)
where is the covariance matrix of the observable data matrix X.
One problem with PCA is that it doesn’t take class labels into account. Therefore projecting using PCA can result in worse class separation, making the classi cation problem more di cult, especially for linear classi ers.
To avoid this, if we have class information, one idea is to separate the data by class and aim to nd the projection that maximize the distance between the means of the class data after pro-jection, while minimizing their variance after projection. This is called linear discriminant analysis (LDA).
Let Ci be the set of observations that have class label i, and i; i be the mean and standard deviations, respectively, of those sets. Assuming that we only have two classes, we then want to nd the value of w that maximizes the equation:
( 1w 2w)T ( 1w 2w) (( 1w)T ( 1w) + ( 2w)T ( 2w))
Which is equivalent to
wT ( 1 2)T ( 1 2)w (wT ( 1T 1 + 2T 2)w)
Show that to maximize we must solve an eigen-decomposition problem, i.e Aw = bw.
In particular what are A and b for this equation.
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(40pts) Dimensionality Reduction via PCA
Download and extract the dataset yalefaces.zip from Blackboard. This dataset has 154 images (N = 154) each of which is a 243x320 image (D = 77760). In order to process this data your script will need to:
Read in the list of les
Create a 154x1600 data matrix such that for each image le
Read in the image as a 2D array (234x320 pixels)
Subsample the image to become a 40x40 pixel image (for processing speed)
Flatten the image to a 1D array (1x1600)
Concatenate this as a row of your data matrix.
Once you have your data matrix, your script should:
Standardizes the data
Reduces the data to 2D using PCA
Graphs the data for visualization
Recall that although you may not use any package ML functions like pca, you may use statistical functions like eig.
Your graph should end up looking similar to Figure 1 (although it may be rotated di erently, de-pending how you ordered things).
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Figure 1: 2D PCA Projection of data
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(20 points) Eigenfaces
Download and extract the dataset yalefaces.zip from Blackboard. This dataset has 154 images (N = 154) each of which is a 243x320 image (D = 77760). In order to process this data your script will need to:
Read in the list of les
Create a 154x1600 data matrix such that for each image le
Read in the image as a 2D array (234x320 pixels)
Subsample the image to become a 40x40 pixel image (for processing speed)
Flatten the image to a 1D array (1x1600)
Concatenate this as a row of your data matrix.
Write a script that:
Imports the data as mentioned above.
Standardizes the data.
Performs PCA on the data (again, although you may not use any package ML functions like pca, you may use statistical functions like eig).
Determines the number of principle components necessary to encode at least 95% of the infor-mation, k.
Visualizes the most important principle component as a 40x40 image (see Figure 2).
Reconstructs the rst person using the primary principle component and then using the k most signi cant eigen-vectors (see Figure 3). For the fun of it maybe even look to see if you can perfectly reconstruct the face if you use all the eigen-vectors!
Your principle eigenface should end up looking similar to Figure 2.
Figure 2: Primary Principle Component
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Your reconstruction should end up looking similar to Figure 3.
Figure 3: Reconstruction of rst person (ID=2)
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Submission
For your submission, upload to Blackboard a single zip le containing:
PDF Writeup
Source Code
readme.txt le
The readme.txt le should contain information on how to run your code to reproduce results for each part of the assignment. Do not include spaces or special characters (other than the underscore character) in your le and directory names. Doing so may break our grading scripts.
The PDF document should contain the following:
Part 1: Your answers to the theory questions.
Part 2: The visualization of the PCA result
Part 3:
Number of principle components needed to represent 95% of information, k.
Visualization of primary principle component
Visualization of the reconstruction of the rst person using
Original image
Single principle component
k principle components.
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