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Homework # 14 SVD

 Homework # 14

1. • Reading from Sauer: The beginning of Section 12.3 dis￾cusses the SVD. Sections 12.4.1 1 12.4.3 discuss applica￾tions of the SVD.
• Reading from Elements of Statistical Learning: Section
14.3.6 discusses K-means.
2. With this assignment you will find the file users-shows.txt.
This file gives a 9985 by 563 matrix, call it A, corresponding to
563 television shows and 9985 television users.. The matrix is
composed of 0’s and 1’s. A 1 means that the user likes the show,
a 0 means they do not. The shows, if you are interested, are
listed in the file shows.txt. The 500th user, let’s call him Alex,
has had his preferences for the first 100 shows removed from
the matrix and replaced with all 0’s. His actual preferences are
in the file Alex.txt. Your goal is to use the svd to suggest 5
shows from the first 100 that you believe Alex would like. You
can then check your answer against his actual preferences. You
should use R’s svd function to compute the SVD of A.
(a) As a warmup to this problem, show following. Given the
(b) Compute the SVD of A and plot the singular values. How
many singular values would accurately approximate this
matrix? (What accurate means here is up to you.)
(c) Use the SVD to reduce the data to two dimensions as fol￾lows. Project the users onto the appropriate two dime￾naional PCA space and plot; do the same for the shows.
Using these projections, suggest five movies for Alex. (This
problem and dataset is taken from a homework in a class
taught by Jure Leskovec in the Stanford CS department.
The dataset was originally produced by Chris Volinsky in
the Columbia CS department.)
1
3. In this problem you will implement K-means on two datasets
(a) Here is a fact I mentioned in class that is essential to the
kmeans algorithm. Suppose you are given N points x
(i)
for
i = 1, 2, . . . , N, with each point in R
n
. Compute the point
m ∈ R
n
that minimizes the sum of squared distances from
To find the m, take the gradient of this expression, set it
to zero, and solve for m. You should find that m is the
mean of the x
(
(b) Write a function MyKmeans(x, K) that accepts a data
matrix X and the number of kmeans K and returns the
solution to the kmeans problem as well as the number of
iterations needed to reach the solution through the kmeans
algorithm discussed in class. (You can check your answer
against R’s kmeans function and, if you like, you can also
include a parameter in MyKmeans that chooses a start￾ing value for the assignments or means.) Explain why the
kmeans algorithm is a descent algorithm.
(c) Apply MyKmeans to the attached dataset synthetic kmeans data.csv
with K = 2. This is an artifical dataset for which the sam￾ple points are in R
. Plot the points of the dataset and the
location of your 2 means at various iterations to see how
the means move to their optimal location.
(d) The attached dataset tumor microarray data.csv comes
from the Elements of Statistical Learning book. Each
row represents a cancer cell. The first column, labeled
cancer, gives the type of the cancer cell (e.g. RENAL,
LEUKEMIA). The rest of the columns are numeric and
give measurement of different proteins in the cell. The
point here is to attempt to distinguish cancer cells by the
level of proteins found in the cell. Perform K-means using
R’s kmeans function. The cluster associated with a given
mean are the sample points assigned to it. Try different K,
and determine if the clusters formed separate the cancers
(e.g. certain cancers are found within certain clusters).
(See Elements of Statistical Learning Table 14.2, which
does this for K = 2.)
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