Exercise 7: K-means Clustering and Principal Component

Programming Exercise 7:
K-means Clustering and Principal Component
Analysis
Machine Learning
Introduction
In this exercise, you will i mplement the K-means clustering algorithm and
apply i t to compress an i mage. In the second part, you will use principal
component analysis to find a l ow-dimensional representation of f ace i mages.
Before starting on the programming exercise, we strongly recommend watching the video l ectures and completing the review questions f or the associated
topics.
To get started with the exercise, you will need to download the starter
code and unzip its contents to the directory where you wish to complete the
exercise. If needed, use the cd command in Octave/MATLAB to change to
this directory before starting this exercise.
You can log into your CougarNet and download MATLAB from this website:
https://uh.edu/software-downloads/index.php.
Files included in this exercise
ex7.m - Octave/MATLAB script for the first exercise on K-means
ex7_pca.m - Octave/MATLAB script for the second exercise on PCA
ex7data1.mat - Example Dataset for PCA
ex7data2.mat - Example Dataset for K-means
ex7faces.mat - Faces Dataset
bird_small.png - Example Image
displayData.m - Displays 2D data stored in a matrix
drawLine.m - Draws a line over an exsiting figure
plotDataPoints.m - Initialization for K-means centroids
plotProgresskMeans.m - Plots each step of K-means as it proceeds
runkMeans.m - Runs the K-means algorithm
[y] pca.m - Perform principal component analysis
[y] projectData.m - Projects a data set into a lower dimensional space
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[y] recoverData.m - Recovers the original data from the projection
[y] findClosestCentroids.m - Find closest centroids (used in K-means)
[y] computeCentroids.m - Compute centroid means (used in K-means)
[y] kMeansInitCentroids.m - Initialization for K-means centroids
y indicates files you will need to complete
Throughout the first part of the exercise, you will be using the script
ex7.m, for the second part you will use ex7 pca.m. These scripts set up the
dataset for the problems and make calls to functions that you will write.
You are only required to modify functions in other files, by following the
instructions in this assignment.
Where to get help
The exercises in this course use Octave1 or MATLAB, a high-level programming language well-suited for numerical computations. If you do not have
Octave or MATLAB installed, please refer to the installation instructions in
the “Environment Setup Instructions” of the course website.
1 K-means Clustering
In this this exercise, you will implement the K-means algorithm and use it
for image compression. You will first start on an example 2D dataset that
1Octave is a free alternative to MATLAB. For the programming exercises, you are free
to use either Octave or MATLAB.
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Files needed to be submit
[1] ML_ex7 – Include all the code (You need to complete pca.m, projectData.m, recoverData.m, findClosestCentroids.m, computeCentroids.m,
and kMeans-InitCentroids.m by yourself)
[2] ex7_report – Directly give the answers of these questions:
(1) The closest centroids class for the first 10 examples
(2) The centroids after the first step of K- means
(3) Show original and reconstructed bird image after you apply K-means
(4) The eigenvalues of the covariance matrix on example dataset
(5) The projection of the first example AND the reconstructed approximation of the first example
will help you gain an intuition of how the K-means algorithm works. After
that, you wil use the K-means algorithm for image compression by reducing
the number of colors that occur in an image to only those that are most
common in that image. You will be using ex7.m for this part of the exercise.
1.1 Implementing K-means
The K-means algorithm is a method to automatically cluster similar data
examples together. Concretely, you are given a training set {x
(1), ..., x(m)}
(where x
(i) ∈ R
n
), and want to group the data into a few cohesive “clusters”.
The intuition behind K-means is an iterative procedure that starts by guessing the initial centroids, and then refines this guess by repeatedly assigning
examples to their closest centroids and then recomputing the centroids based
on the assignments.
The K-means algorithm is as follows:
% Initialize centroids
centroids = kMeansInitCentroids(X, K);
for iter = 1:iterations
% Cluster assignment step: Assign each data point to the
% closest centroid. idx(i) corresponds to cˆ(i), the index
% of the centroid assigned to example i
idx = findClosestCentroids(X, centroids);
% Move centroid step: Compute means based on centroid
% assignments
centroids = computeMeans(X, idx, K);
end
The inner-loop of the algorithm repeatedly carries out two steps: (i) Assigning each training example x
(i)
to its closest centroid, and (ii) Recomputing the mean of each centroid using the points assigned to it. The K-means
algorithm will always converge to some final set of means for the centroids.
Note that the converged solution may not always be ideal and depends on the
initial setting of the centroids. Therefore, in practice the K-means algorithm
is usually run a few times with different random initializations. One way to
choose between these different solutions from different random initializations
is to choose the one with the lowest cost function value (distortion).
You will implement the two phases of the K-means algorithm separately
in the next sections.
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1.1.1 Finding closest centroids
In the “cluster assignment” phase of the K-means algorithm, the algorithm
assigns every training example x
(i)
to its closest centroid, given the current
positions of centroids. Specifically, for every example i we set
c
(i)
:= j that minimizes ||x
(i) − µj
||2
,
where c
(i)
is the index of the centroid that is closest to x
(i)
, and µj
is the
position (value) of the j’th centroid. Note that c
(i)
corresponds to idx(i) in
the starter code.
Your task is to complete the code in findClosestCentroids.m. This
function takes the data matrix X and the locations of all centroids inside
centroids and should output a one-dimensional array idx that holds the
index (a value in {1, ..., K}, where K is total number of centroids) of the
closest centroid to every training example.
You can implement this using a loop over every training example and
every centroid.
Once you have completed the code in findClosestCentroids.m, the
script ex7.m will run your code and you should see the output [1 3 2]
corresponding to the centroid assignments for the first 3 examples.
1.1.2 Computing centroid means
Given assignments of every point to a centroid, the second phase of the
algorithm recomputes, for each centroid, the mean of the points that were
assigned to it. Specifically, for every centroid k we set
µk :=
1
|Ck|
X
i∈Ck
x
(i)
where Ck is the set of examples that are assigned to centroid k. Concretely,
if two examples say x
(3) and x
(5) are assigned to centroid k = 2, then you
should update µ2 =
1
2
(x
(3) + x
(5)).
You should now complete the code in computeCentroids.m. You can
implement this function using a loop over the centroids. You can also use a
loop over the examples; but if you can use a vectorized implementation that
does not use such a loop, your code may run faster.
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Once you have completed the code in computeCentroids.m, the script
ex7.m will run your code and output the centroids after the first step of Kmeans.
1.2 K-means on example dataset
−1 0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
Iteration number 10
Figure 1: The expected output.
After you have completed the two functions (findClosestCentroids and
computeCentroids), the next step in ex7.m will run the K-means algorithm
on a toy 2D dataset to help you understand how K-means works. Your
functions are called from inside the runKmeans.m script. We encourage you
to take a look at the function to understand how it works. Notice that the
code calls the two functions you implemented in a loop.
When you run the next step, the K-means code will produce a visualization that steps you through the progress of the algorithm at each iteration.
Press enter multiple times to see how each step of the K-means algorithm
changes the centroids and cluster assignments. At the end, your figure should
look as the one displayed in Figure 1.
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1.3 Random initialization
The initial assignments of centroids for the example dataset in ex7.m were
designed so that you will see the same figure as in Figure 1. In practice, a
good strategy for initializing the centroids is to select random examples from
the training set.
In this part of the exercise, you should complete the function kMeansInitCentroids.m
with the following code:
% Initialize the centroids to be random examples
% Randomly reorder the indices of examples
randidx = randperm(size(X, 1));
% Take the first K examples as centroids
centroids = X(randidx(1:K), :);
The code above first randomly permutes the indices of the examples (using randperm). Then, it selects the first K examples based on the random
permutation of the indices. This allows the examples to be selected at random without the risk of selecting the same example twice.
1.4 Image compression with K-means
Figure 2: The original 128x128 image.
In this exercise, you will apply K-means to image compression. In a
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straightforward 24-bit color representation of an image,2
each pixel is represented as three 8-bit unsigned integers (ranging from 0 to 255) that specify
the red, green and blue intensity values. This encoding is often refered to as
the RGB encoding. Our image contains thousands of colors, and in this part
of the exercise, you will reduce the number of colors to 16 colors.
By making this reduction, it is possible to represent (compress) the photo
in an efficient way. Specifically, you only need to store the RGB values of
the 16 selected colors, and for each pixel in the image you now need to only
store the index of the color at that location (where only 4 bits are necessary
to represent 16 possibilities).
In this exercise, you will use the K-means algorithm to select the 16 colors
that will be used to represent the compressed image. Concretely, you will
treat every pixel in the original image as a data example and use the K-means
algorithm to find the 16 colors that best group (cluster) the pixels in the 3-
dimensional RGB space. Once you have computed the cluster centroids on
the image, you will then use the 16 colors to replace the pixels in the original
image.
1.4.1 K-means on pixels
In Octave/MATLAB, images can be read in as follows:
% Load 128x128 color image (bird small.png)
A = imread('bird small.png');
% You will need to have installed the image package to used
% imread. If you do not have the image package installed, you
% should instead change the following line to
%
% load('bird small.mat'); % Loads the image into the variable A
This creates a three-dimensional matrix A whose first two indices identify
a pixel position and whose last index represents red, green, or blue. For
example, A(50, 33, 3) gives the blue intensity of the pixel at row 50 and
column 33.
The code inside ex7.m first loads the image, and then reshapes it to create
an m × 3 matrix of pixel colors (where m = 16384 = 128 × 128), and calls
your K-means function on it.
After finding the top K = 16 colors to represent the image, you can now
2The provided photo used in this exercise belongs to Frank Wouters and is used with
his permission.
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assign each pixel position to its closest centroid using the findClosestCentroids
function. This allows you to represent the original image using the centroid
assignments of each pixel. Notice that you have significantly reduced the
number of bits that are required to describe the image. The original image
required 24 bits for each one of the 128×128 pixel locations, resulting in total
size of 128 × 128 × 24 = 393, 216 bits. The new representation requires some
overhead storage in form of a dictionary of 16 colors, each of which require
24 bits, but the image itself then only requires 4 bits per pixel location. The
final number of bits used is therefore 16 × 24 + 128 × 128 × 4 = 65, 920 bits,
which corresponds to compressing the original image by about a factor of 6.
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Figure 3: Original and reconstructed image (when using K-means to compress the image).
Finally, you can view the effects of the compression by reconstructing the
image based only on the centroid assignments. Specifically, you can replace
each pixel location with the mean of the centroid assigned to it. Figure 3
shows the reconstruction we obtained. Even though the resulting image retains most of the characteristics of the original, we also see some compression
artifacts.
1.5 Optional (ungraded) exercise: Use your own image
In this exercise, modify the code we have supplied to run on one of your
own images. Note that if your image is very large, then K-means can take a
long time to run. Therefore, we recommend that you resize your images to
managable sizes before running the code. You can also try to vary K to see
the effects on the compression.
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2 Principal Component Analysis
In this exercise, you will use principal component analysis (PCA) to perform
dimensionality reduction. You will first experiment with an example 2D
dataset to get intuition on how PCA works, and then use it on a bigger
dataset of 5000 face image dataset.
The provided script, ex7 pca.m, will help you step through the first half
of the exercise.
2.1 Example Dataset
To help you understand how PCA works, you will first start with a 2D dataset
which has one direction of large variation and one of smaller variation. The
script ex7 pca.m will plot the training data (Figure 4). In this part of the
exercise, you will visualize what happens when you use PCA to reduce the
data from 2D to 1D. In practice, you might want to reduce data from 256 to
50 dimensions, say; but using lower dimensional data in this example allows
us to visualize the algorithms better.
1 2 3 4 5 6
2
3
4
5
6
7
8
Figure 4: Example Dataset 1
2.2 Implementing PCA
In this part of the exercise, you will implement PCA. PCA consists of two
computational steps: First, you compute the covariance matrix of the data.
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Then, you use Octave/MATLAB’s SVD function to compute the eigenvectors U1, U2, . . . , Un. These will correspond to the principal components of
variation in the data.
Before using PCA, it is important to first normalize the data by subtracting the mean value of each feature from the dataset, and scaling each dimension so that they are in the same range. In the provided script ex7 pca.m,
this normalization has been performed for you using the featureNormalize
function.
After normalizing the data, you can run PCA to compute the principal
components. You task is to complete the code in pca.m to compute the principal components of the dataset. First, you should compute the covariance
matrix of the data, which is given by:
Σ = 1
m
X
T X
where X is the data matrix with examples in rows, and m is the number of
examples. Note that Σ is a n × n matrix and not the summation operator.
After computing the covariance matrix, you can run SVD on it to compute
the principal components. In Octave/MATLAB, you can run SVD with the
following command: [U, S, V] = svd(Sigma), where U will contain the
principal components and S will contain a diagonal matrix.
1 2 3 4 5 6
2
3
4
5
6
7
8
Figure 5: Computed eigenvectors of the dataset
Once you have completed pca.m, the ex7 pca.m script will run PCA on
the example dataset and plot the corresponding principal components found
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(Figure 5). The script will also output the top principal component (eigenvector) found, and you should expect to see an output of about [-0.707
-0.707]. (It is possible that Octave/MATLAB may instead output the negative of this, since U1 and −U1 are equally valid choices for the first principal
component.)
2.3 Dimensionality Reduction with PCA
After computing the principal components, you can use them to reduce the
feature dimension of your dataset by projecting each example onto a lower
dimensional space, x
(i) → z
(i)
(e.g., projecting the data from 2D to 1D). In
this part of the exercise, you will use the eigenvectors returned by PCA and
project the example dataset into a 1-dimensional space.
In practice, if you were using a learning algorithm such as linear regression
or perhaps neural networks, you could now use the projected data instead
of the original data. By using the projected data, you can train your model
faster as there are less dimensions in the input.
2.3.1 Projecting the data onto the principal components
You should now complete the code in projectData.m. Specifically, you are
given a dataset X, the principal components U, and the desired number of
dimensions to reduce to K. You should project each example in X onto the
top K components in U. Note that the top K components in U are given by
the first K columns of U, that is U reduce = U(:, 1:K).
Once you have completed the code in projectData.m, ex7 pca.m will
project the first example onto the first dimension and you should see a value
of about 1.481 (or possibly -1.481, if you got −U1 instead of U1).
2.3.2 Reconstructing an approximation of the data
After projecting the data onto the lower dimensional space, you can approximately recover the data by projecting them back onto the original high
dimensional space. Your task is to complete recoverData.m to project each
example in Z back onto the original space and return the recovered approximation in X rec.
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Once you have completed the code in recoverData.m, ex7 pca.m will
recover an approximation of the first example and you should see a value of
about [-1.047 -1.047].
2.3.3 Visualizing the projections
−4 −3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
2
3
Figure 6: The normalized and projected data after PCA.
After completing both projectData and recoverData, ex7 pca.m will
now perform both the projection and approximate reconstruction to show
how the projection affects the data. In Figure 6, the original data points are
indicated with the blue circles, while the projected data points are indicated
with the red circles. The projection effectively only retains the information
in the direction given by U1.
2.4 Face Image Dataset
In this part of the exercise, you will run PCA on face images to see how it
can be used in practice for dimension reduction. The dataset ex7faces.mat
contains a dataset3 X of face images, each 32 × 32 in grayscale. Each row
of X corresponds to one face image (a row vector of length 1024). The next
3This dataset was based on a cropped version of the labeled faces in the wild dataset.
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step in ex7 pca.m will load and visualize the first 100 of these face images
(Figure 7).
Figure 7: Faces dataset
2.4.1 PCA on Faces
To run PCA on the face dataset, we first normalize the dataset by subtracting
the mean of each feature from the data matrix X. The script ex7 pca.m will
do this for you and then run your PCA code. After running PCA, you will
obtain the principal components of the dataset. Notice that each principal
component in U (each row) is a vector of length n (where for the face dataset,
n = 1024). It turns out that we can visualize these principal components by
reshaping each of them into a 32 × 32 matrix that corresponds to the pixels
in the original dataset. The script ex7 pca.m displays the first 36 principal
components that describe the largest variations (Figure 8). If you want, you
can also change the code to display more principal components to see how
they capture more and more details.
2.4.2 Dimensionality Reduction
Now that you have computed the principal components for the face dataset,
you can use it to reduce the dimension of the face dataset. This allows you to
use your learning algorithm with a smaller input size (e.g., 100 dimensions)
instead of the original 1024 dimensions. This can help speed up your learning
algorithm.
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Figure 8: Principal components on the face dataset
Figure 9: Original images of faces and ones reconstructed from only the top
100 principal components.
The next part in ex7 pca.m will project the face dataset onto only the
first 100 principal components. Concretely, each face image is now described
by a vector z
(i) ∈ R
100
.
To understand what is lost in the dimension reduction, you can recover
the data using only the projected dataset. In ex7 pca.m, an approximate
recovery of the data is performed and the original and projected face images
are displayed side by side (Figure 9). From the reconstruction, you can observe that the general structure and appearance of the face are kept while
the fine details are lost. This is a remarkable reduction (more than 10×) in
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the dataset size that can help speed up your learning algorithm significantly.
For example, if you were training a neural network to perform person recognition (gven a face image, predict the identitfy of the person), you can use
the dimension reduced input of only a 100 dimensions instead of the original
pixels.
2.5 Optional (ungraded) exercise: PCA for visualization
Figure 10: Original data in 3D
In the earlier K-means image compression exercise, you used the K-means
algorithm in the 3-dimensional RGB space. In the last part of the ex7 pca.m
script, we have provided code to visualize the final pixel assignments in this
3D space using the scatter3 function. Each data point is colored according
to the cluster it has been assigned to. You can drag your mouse on the figure
to rotate and inspect this data in 3 dimensions.
It turns out that visualizing datasets in 3 dimensions or greater can be
cumbersome. Therefore, it is often desirable to only display the data in 2D
even at the cost of losing some information. In practice, PCA is often used to
reduce the dimensionality of data for visualization purposes. In the next part
of ex7 pca.m, the script will apply your implementation of PCA to the 3-
dimensional data to reduce it to 2 dimensions and visualize the result in a 2D
scatter plot. The PCA projection can be thought of as a rotation that selects
the view that maximizes the spread of the data, which often corresponds to
the “best” view.
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Figure 11: 2D visualization produced using PCA
Submission and Grading
After completing various parts of the assignment, be sure to use the submit
to blackboard. The following is a breakdown of how each part of this
exercise is scored.
Part Submitted File Points
The closest centroids class for the first 30 points
10 examples
The centroids after the first step of Kmeans
Show original and reconstructed bird
image after you apply K-means
findClosestCentroids.m
computeCentroids.m
kMeansInitCentroids.m
25 points
5 points
The eigenvalues of the covariance 20 points
matrix on example dataset
The projection of the first example
AND the reconstructed approximation of the first example
pca.m
projectData.m
recoverData.m
20 points
Total Points 100 points
You are allowed to submit your solutions multiple times, and we will take
only the highest score into consideration.
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