## Description

Problem 1 – Clustering with κ-means We will consider the UCI ML Breast Cancer

Wisconsin dataset. Features are computed from a digitized image of a fine needle aspirate

(FNA) of a breast mass. They describe characteristics of the cell nuclei present in the image.

You can download the dataset using the function:

sklearn.datasets.load breast cancer

Unless specified otherwise, questions in Problem 1 below refer to the UCI ML Breast Cancer

Wisconsin dataset when the word “dataset” is used.

1. (5 points) Implement κ-means yourself. Your function should take in an array containing

a dataset and a value of κ, and return the cluster centroids along with the cluster

assignment for each data point. You may choose the initialization heuristic of your

choice among the two we saw in class. Hand-in the code for full credit. For this

question, you should not rely on any library other than numPy in Python.

Fall 2022 – Introduction to ML Assignment 1 – Page 2 of 2 Sep 22

2. (1 point) Run the κ-means algorithm for values of κ varying between 2 and 7, at increments of 1. Justify in your answer which data you passed as the input to the κ-means

algorithm.

3. (2 points) Plot the distortion achieved by κ-means for values of κ varying between 2

and 7, at increments of 1. Hand-in the code and figure output for full credit. For this

question, you may rely on plotting libraries such as matplotlib.

4. (1 point) If you had to pick one value of κ, which value would you pick? Justify your

choice.

Problem 2 – Lack of optimality of κ-means

1. (3 points) Construct an analytical demonstration that κ-means might converge to a

solution that is not globally optimal. Hint: consider the case where κ = 2 and the dataset

is made up of 4 points in R as follows: x(1) = 1, x(2) = 2, x(3) = 3, x(4) = 4. Initialize

κ-means with the centroids µ1 = 2 and µ2 = 4. Note: you may assume that if a point

x(i) is equally distant to multiple centroids µk, the point will be assigned to the centroid

whose index is smallest, i.e., k with the smallest value for k ∈ arg mink “x(i) − µk”2.

Problem 3 – SVD This problem will help you review background required to understand

an upcoming lecture video on PCA, which involves singular value decomposition (SVD).

Consider the following matrix:

A =

!

1 2 1

2 3 1

”

All following questions should be answered “by hand”, i.e., you should derive the results

analytically—not using a computer program. You may only use your computer as a simple

calculator: this means you can for instance use the computer to multiply two real numbers,

but not to compute the matrices asked for in the question using MATLAB. In the following,

we use the notation UΣV ⊤ to denote the SVD of matrix A

1. (1 point) Show that the matrix A is of rank 2.

2. (2 points) Show that the singular values of the matrix A are σ1 = #10 + √97 and

σ2 = #10 − √97

3. (5 points) Derive the matrices U and V for the SVD of matrix A.

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