Assignment 1 Dimensionality Reduction Solution

$30.00

Description

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 243×320 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

    1. (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

    1. Compute the information gain for each feature. You could standardize the data overall, although it won’t make a di erence. (13pts).

    1. Which feature is more discriminating based on results in Part (a) (2pt)?

  1. (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.

3

  • (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 243×320 image (D = 77760). In order to process this data your script will need to:

  1. Read in the list of les

  1. Create a 154×1600 data matrix such that for each image le

    1. Read in the image as a 2D array (234×320 pixels)

    1. Subsample the image to become a 40×40 pixel image (for processing speed)

    1. Flatten the image to a 1D array (1×1600)

    1. Concatenate this as a row of your data matrix.

Once you have your data matrix, your script should:

  1. Standardizes the data

  1. Reduces the data to 2D using PCA

  1. 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).

4

Figure 1: 2D PCA Projection of data

5

  • (20 points) Eigenfaces

Download and extract the dataset yalefaces.zip from Blackboard. This dataset has 154 images (N = 154) each of which is a 243×320 image (D = 77760). In order to process this data your script will need to:

  1. Read in the list of les

  1. Create a 154×1600 data matrix such that for each image le

    1. Read in the image as a 2D array (234×320 pixels)

    1. Subsample the image to become a 40×40 pixel image (for processing speed)

    1. Flatten the image to a 1D array (1×1600)

    1. Concatenate this as a row of your data matrix.

Write a script that:

  1. Imports the data as mentioned above.

  1. Standardizes the data.

  1. Performs PCA on the data (again, although you may not use any package ML functions like pca, you may use statistical functions like eig).

  1. Determines the number of principle components necessary to encode at least 95% of the infor-mation, k.

  1. Visualizes the most important principle component as a 40×40 image (see Figure 2).

  1. 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:

  1. PDF Writeup

  1. Source Code

  1. 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:

  1. Part 1: Your answers to the theory questions.

  1. Part 2: The visualization of the PCA result

  1. Part 3:

    1. Number of principle components needed to represent 95% of information, k.

    1. Visualization of primary principle component

    1. Visualization of the reconstruction of the rst person using

      1. Original image

      1. Single principle component

      1. k principle components.

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