[SOLVED] Cs550 problem set 2

In this problem we will explore the relationship between two of the most popular dimensionality reduction techniques, SVD and PCA at a basic conceptual level. Before we
proceed with the question itself, let us briefly recap the SVD and PCA techniques and a
few important observations:• First, recall that the eigenvalue decomposition of a real, symmetric, and square matrix
B (of size d × d) can be written as the following product:
B = QΛQ
T
where Λ = diag(λ1, · · · , λd) contains the eigenvalues of B (which are always real) along
its main diagonal and Q is an orthogonal matrix containing the eigenvectors of B as its
columns.• Principal Component Analysis (PCA): Given a data matrix M (of size p × q), PCA
involves the computation of the eigenvectors of MMT or MTM. The matrix of these
eigenvectors can be thought of as a rigid rotation in a high dimensional space. When you
apply this transformation to the original data, the axis corresponding to the principal
eigenvector is the one along which the points are most spread out. More precisely, this
axis is the one along which the variance of the data is maximized. Put another way, the
points can best be viewed as lying along this axis, with small deviations from this axis.Likewise, the axis corresponding to the second eigenvector (the eigenvector corresponding
to the second-largest eigenvalue) is the axis along which the variance of distances from
the first axis is greatest, and so on.• Singular Value Decomposition (SVD): SVD involves the decomposition of a data matrix
M (of size p × q) into a product: UΣV
T where U (of size p × r) and V (of size q × r)
are column-orthonormal matrices1 and Σ (of size r × r) is a diagonal matrix. The entries
along the diagonal of Σ are referred to as singular values of M. The key to understanding
what SVD offers is in viewing the r columns of U, Σ, and V as representing concepts that
are hidden in the original matrix M.For answering the questions below, let us define a matrix M (of size p × q) and let us
assume this matrix corresponds to a dataset with p data points and q dimensions.
(a) [5pt] Are the matrices MMT and MTM symmetric, square and real? Explain.
(b) [5pt] Prove that the eigenvalues of MMT are the same as that of MTM. Are their
eigenvectors the same?(c) [5pt] Given that we now understand certain properties of MTM, write an expression
for MTM in terms of Q, QT and Λ where Λ = diag(λ1, …, λd) contains the eigenvalues of
MTM along its main diagonal and Q is an orthogonal matrix containing the eigenvectors
of MTM as its columns. (Hint: Check the definition of eigenvalue decomposition provided
in the beginning of the question to see if it is applicable.)(d) [5pt] SVD decomposes the matrix M into the product UΣV
T where U and V are
column-orthonormal and Σ is a diagonal matrix. Given that M = UΣV
T
, write a simplified expression for MTM in terms of V, V T and Σ.(e) In this question, let us experimentally test if SVD decomposition of M actually provides us the eigenvectors (PCA dimensions) of MTM. We strongly recommend students
to use Python and suggested functions for this exercise2
.
Initialize matrix M as follows:
M =




1 2
2 1
3 4
4 3



• (e)(a) [5pt] Compute the SVD of M (Use scipy.linalg.svd function in Python and set the
argument full matrices to False). The function returns values corresponding to U, Σ
and V
T
. What are the values returned for U, Σ and V
T
?Note: Make sure that the first
element of the returned array Σ has a greater value than the second element.
• (e)(b) [5pt] Compute the eigenvalue decomposition of MTM (Use scipy.linalg.eigh function in Python). The function returns two parameters: a list of eigenvalues (let us call
1A matrix U ∈ R
p×q
is column-orthonormal if and only if U
TU = I where I denotes the identity matrix
2Other implementations of SVD and PCA might give slightly different results.Besides, you will just need
fewer than five python commands to answer this entire question
this list Evals) and a matrix whose columns correspond to the eigenvectors of the respective eigenvalues (let us call this matrix Evecs).Sort the list Evals in descending order
such that the largest eigenvalue appears first in the list. Also, re-arrange the columns in
Evecs such that the eigenvector corresponding to the largest eigenvalue appears in the
first column of Evecs. What are the values of Evals and Evecs (after the sorting and
re-arranging process)?• (e)(c) [5pt] Based on the experiment and your derivations in part (c) and (d), do you see
any correspondence between V produced by SVD and the matrix of eigenvectors Evecs
(after the sorting and re-arranging process) produced by eigenvalue decomposition? If so,
what is it? (Note: The function scipy.linalg.svd returns V
T
, not V .)• (e)(d) [5pt] Based on the experiment and the expressions obtained in part (c) and part
(d) for MTM, what is the relationship (if any) between the eigenvalues of MTM and the
singular values of M? Explain.Note: The entries along the diagonal of Σ (part (d)) are referred to as singular values of
M. The eigenvalues of MTM are captured by the diagonal elements in Λ (part (c)).What to submit:
(i) Written solutions to questions 1(a) to 1(e) with explanations wherever required.
(ii) Include the code as a single Python file as question 1.py.Let the matrix of the Web M be an n-by-n matrix, where n is the number of Web pages.
The entry mij in row i and column j is 0, unless there is an arc from node (page) j to
node i. In that case, the value of mij is 1/k, where k is the number of arcs (links) out of
node j. Notice that if node j has k > 0 arcs out, then column j has k values of 1/k and
the rest 0’s. If node j is a dead end (i.e., it has zero arcs out), then column j is all 0’s.
Let r = [r1, r2, …, rn]T be (an estimate of) the PageRank vector; that is, ri
is the estimate
of the PageRank of node i. Define w(r) to be the sum of the components of r; that is
w(r) = Pn
j=1 ri
.
In one iteration of the PageRank algorithm, we compute the next estimate r
0 of the
PageRank as: r
0 = Mr. Specifically, for each i we compute r
0
i =
Pn
j=1 Mijrj
.(a) [5pt] Suppose the Web has no dead ends. Prove that w(r
0
) = w(r).
(b) [5pt] Suppose there are still no dead ends, but we use a teleportation probability
of 1 − β, where 0 < β < 1. The expression for the next estimate of ri becomes r
0
i =
β
Pn
j=1 Mijrj + (1 − β)/n. Under what circumstances will w(r
0
) = w(r)? Prove your
conclusion.(c) Now, let us assume a teleportation probability of 1 − β in addition to the fact that
there are one or more dead ends. Call a node “dead” if it is a dead end and “live” if not.
Assume w(r) = 1. At each iteration, we will distribute equally to each node the sum of:
• 1. (1 − β)rj
if node j is live.
• 2. rj
if node j is dead.
3
• (c)(a) [5pt] Write the equation for r
0
i
in terms of β, M, and r.
• (c)(b) [5pt] Then, prove that w(r
0
) is also 1.
What to submit:
(i) Proof of 2(a);
(ii) Condition for w(r
0
) = w(r) and Proof of 2(b);
(iii) Equation for r
0
i
and Proof of 2(c).In this problem, you will learn how to implement the PageRank algorithm. You will be
experimenting with the provided graph (assume graph has no dead-ends), which is stores
in the graph.txt file.It has n = 100 nodes (numbered 1, 2, …, 100), and m = 1024 edges, 100 of which form a
directed cycle (through all the nodes) which ensures that the graph is connected. It is
easy to see that the existence of such a cycle ensures that there are no dead ends in the
graph. There may be multiple edges between a pair of nodes, your program should handle
these instead of ignoring them. The first column in graph.txt refers to the source node,
and the second column refers to the destination node.Assume the directed graph G = (V, E) has n nodes (numbered 1, 2, …, n) and m edges,
all nodes have positive out-degree, and M = [Mji]n×n is an n × n matrix as defined in
class such that for any i, j ∈ [1, n]:
Mji =
( 1
deg(i)
, if (i → j) ∈ E
0 otherwiseHere, deg(i) is the number of outgoing edges of node i in G. By the definition of PageRank,
assuming 1 − β to be the teleport probability, and denoting the PageRank vector by the
column vector r, we have the following equation:
r =
1 − β
n
1 + βMr
where 1 is the n × 1 vector with all entries equal to 1.Based on this equation, the iterative procedure to compute PageRank works as follows:
1. Initialize: r
(0) =
1
n
1;
2. For i from 1 to k, iterate: r
(i) =
1−β
n
1 + βMr
(i−1)
.Run the aforementioned iterative process for 40 iterations (assuming β = 0.8) and obtain
the PageRank vector r. Compute the following:
(a) [10pt] List the top 5 node IDs with the highest PageRank scores.
(b) [10pt] List the bottom 5 node IDs with the lowest PageRank scores.
What to submit:
(i) List 5 node IDs with the highest and least PageRank scores in your writeup.
(ii) Include your code as a single source code file such as question 2.py.Note: This problem requires substantial computing time. Don’t start it at the last
minute.This problem will help you understand the nitty gritty details of implementing clustering
algorithms on Hadoop. In addition, this problem will also help you understand the impact
of using various initialization strategies in practice. Let us say we have a set X of n data
points in the d-dimensional space R
d
. Given the number of clusters k and the set of k
centroids C, we now proceed to define the distance metric and the corresponding cost
function that we minimize.Euclidean distance: Given two points A and B in d dimensional space such that
A = [a1, a2…ad] and B = [b1, b2…bd], the Euclidean distance between A and B is defined
as:
ka − bk =
vuutX
d
i=1
(ai − bi)
2The corresponding cost function φ that is minimized when we assign points to clusters
using the Euclidean distance metric is given by:
φ =
X
x∈X
min
c∈C
kx − ck
2Iterative k-Means Algorithm: We learned the basic k-Means algorithm in class which is
as follows: k centroids are initialized, each point is assigned to the nearest centroid and
the centroids are recomputed based on the assignments of points to clusters. In practice,
the above steps are run for several iterations. We present the resulting iterative version
of k-Means in Algorithm 1.Iterative k-Means clustering on Hadoop: Implement iterative k-means using MapReduce
where a single step of MapReduce completes one iteration of the k-means algorithm. So,
to run k-means for i iterations, you will have to run a sequence of i MapReduce jobs.
Please use our provided dataset hw2-q4-kmeans.zip for this problem. The zip has 4 files:• 1. data.txt contains the dataset which has 4601 rows and 58 columns. Each row is
a document represented as a 58 dimensional vector of features. Each component in
the vector represents the importance of a word in the document.• 2. c1.txt contains k initial cluster centroids. These centroids were chosen by selecting k = 10 random points from the input data.
• 3. c2.txt contains initial cluster centroids which are as far apart as possible. (You
can do this by choosing the 1st centroid c1 randomly, and then finding the point c2
that is farthest from c1, then selecting c3 which is farthest from c1 and c2, and so
on).• 4. vocab.txt is the vocabulary document that contains the words, this file is only
for your reference and you do not need it to do the experiment.
Set number of iterations (MAX ITER) to 20 and number of clusters k to 10 for all the
experiments carried out in this question.Hint about job chaining:
We need to run a sequence of Hadoop jobs where the output of one job will be the input
for the next one. There are multiple ways to do this and you are free to use any method
you are comfortable with. One simple way to handle such a multistage job is to configure
the output path of the first job to be the input path of the second and so on.The following pseudo code demonstrates job chaining.
var inputDir
var outputDir
var centroidDir
for i in number-of-iterations (
Configure job here with all params
Set job input directory = inputDir
Set job output directory = outputDir + i
Run job
centroidDir = outputDir + i
)You will also need to share the location of the centroid file with the mapper. There are
many ways to do this and you can use any method you find suitable. One way is to use
the Hadoop Configuration object. You can set it as a property in the Configuration object
and retrieve the property value in the Mapper setup function.For more details see :
• http://hadoop.apache.org/docs/r1.0.4/api/org/apache/hadoop/conf/Configuration
.html#set(java.lang.String,java.lang.String)
• http://hadoop.apache.org/docs/r1.0.4/api/org/apache/hadoop/conf/Configuration
.html#get(java.lang.String)
• http://hadoop.apache.org/docs/r1.0.4/api/org/apache/hadoop/mapreduce/Mapper
.html#setup(org.apache.hadoop.mapreduce.Mapper.Context)(a) [10pt] Using the Euclidean distance as the distance measure, compute the cost function
φ(i) for every iteration i. This means that, for your first MapReduce job iteration, you’ll
be computing the cost function using the initial centroids located in one of the two text
files. Run the k-means on data.txt using c1.txt and c2.txt. Generate a graph where
you plot the cost function φ(i) as a function of the number of iterations i = 1, 2…20 for
c1.txt and also for c2.txt.(Hint: Note that you do not need to write a separate MapReduce job to compute φ(i).
You can just incorporate the computation of φ(i) into the Mapper/Reducer.)(b) [10pt] What is the percentage change in cost after 10 iterations of the k-Means algorithm when the cluster centroids are initialized using c1.txt vs. c2.txt? Is random
initialization of k-means using c1.txt better than initialization using c2.txt in terms of
cost φ(i)? Explain your reasoning.What to submit:
(i) A plot of cost vs. iteration for two initialization strategies for 4(a)
(ii) Percentage improvement values and your explanation for 4(b)
(iii) The code named as question 4.py or question 4.java.