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NumPy, Computational Linear Algebra, and Word2Vec

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Assignment 1: NumPy, Computational Linear Algebra, and
Word2Vec
LING 380/780
Neural Network Models of Linguistic Structure

Total Points: 100
This assignment will introduce you to the manipulation of vectors and matrices with Python
and give you the opportunity to apply these skills to Word2Vec embeddings by implementing a function that computes analogies. For the assignment, please download all the files in
the homework/homework 1 folder in the Files section of Canvas. You will complete Python
programming exercises by filling out function definitions in the numpy exercise.py and
word2vec exercise.py files, and you will put written answers in a Markdown1 file called
assignment1.md.
1 Linear Algebra and NumPy Exercises
In the first part of the assignment, you will practice using NumPy, a Python library that
efficiently implements linear algebra operations.
1.1 Installation
To obtain the software necessary for this assignment, please download and install the
Anaconda distribution of Python, available at https://www.anaconda.com/. If you do
not have Anaconda installed already, you should choose the latest version of the Individual
Edition, which comes with Python 3.8. Anaconda comes with a number of Python packages
that are essential to working with neural networks, as well as the conda package manager.
If you do not have enough disk space for Anaconda, you can choose to install Miniconda
instead, available at https://docs.conda.io/en/latest/miniconda.html. Miniconda is
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https://www.markdownguide.org/basic-syntax/
1
a version of Anaconda that only includes the conda package manager. If you choose to use
Miniconda or an existing Python installation,2 please ensure that you have the following
installed.
• Python 3.5 or a later version
• The pip package manager or conda package manager. If you don’t have either package
manager, please install pip by following the instructions at https://pip.pypa.io/
en/stable/installation/.
• The NumPy package. If you don’t have NumPy installed, please follow the instructions at https://numpy.org/install/.
1.2 Basic Operations (3 Points Each)
In the following exercises, you will read snippets of code and describe what they do in
plain English. You may choose to run the code snippets in the Python console, a Python
script, or a Jupyter Notebook.3 You may also consult the NumPy documentation at
https://numpy.org/doc/stable/. Each code snippet assumes that all previous code
snippets have already been run. Therefore, you must run the code snippets in the same
order as they appear in the instructions.
To begin, please import the NumPy package, as follows.
1 import numpy as np
By convention, we typically refer to the NumPy package using the alias np.
Problem 1. The heart of NumPy is the array, a data type that represents vectors and
matrices. Please create some arrays using the following code.
1 a = np . array (1)
2 b = np . array ([1 , 2 , 3])
3 c = np . array ([[1 , 2 , 3] , [4 , 5 , 6]])
4 d = np . array ([[[1 , 2 , 3] , [4 , 5 , 6]] ,
5 [[7 , 8 , 9] , [10 , 11 , 12]]])
What mathematical objects are represented by a, b, c, and d?
Problem 2. An important property of an array is its shape. Please run the following code
in order to inspect the shape of our arrays.
2This is not recommended unless you have an existing Python or Anaconda setup that you are already
accustomed to using. If you are new to Python, you should install Anaconda, even if your computer (Mac
OS, Linux, and other Unix-based operating systems) has a pre-installed version of Python.
3
https://jupyter.org/
2
1 print ( a . shape )
2 print ( b . shape )
3 print ( c . shape )
4 print ( d . shape )
What is the .shape of an array?
Problem 3. What do the following lines of code do? In your answer, refer to the lines by
their line number (line 1, line 2, etc.).
1 c [0]
2 c [: , 0]
3 c [ -1]
4 c [1:]
5 c [: , np . newaxis ]
Problem 4. Please run the following lines of code.
1 print ( c + c )
2 print ( c – c )
3 print ( c * c )
4 print ( c / c )
What do +, -, *, and / do?
Problem 5. Please run the following code.
1 print (5 + c )
2 print ( a + c )
3 print ( b + c )
What happens when you add a scalar or a vector to a matrix?
Problem 6. Please run the following code.
1 print ( c . T )
2 print ( c @ c . T )
What does .T do? What does @ do?
Problem 7. Please run the following code.
1 # Create a random array of shape (2 , 3 , 4)
2 e = np . random . randint ( -10 , high =10 , size =(2 , 3 , 4) )
3 print ( d @ e )
4 print (( d @ e ) . shape )
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What does @ do when its arguments are 3-dimensional or higher-dimensional arrays?
Problem 8. Please run the following code.
1 print ( c .sum () )
2 print ( c .sum( axis =0) )
3 print ( c .sum( axis =1) )
4 print ( c .sum( axis =1 , keepdims = True ) )
What does .sum() do? What do the axis and keepdims keyword parameters do?
1.3 Complex Operations (6 Points Each)
Next, you will implement some array operations that are frequently used in machine learning. In each of the problems below, you must implement the operation described
using only one line of code. Your one line of code cannot exceed 76 characters in width,
excluding indentations. For each problem, you must fill out one of the functions in the
numpy exercise.py file by replacing pass with your own code. Please read the docstring4
of each function for guidance on what each function should do. You should consult the
NumPy documentation and look for operations that will help you implement the functions.
Problem 9. Please implement the sigmoid function, which applies sigmoid to each item
in an array.
Problem 10. Please implement the zero center function, which takes a matrix and
subtracts the mean value from each row.
Problem 11. Please implement the even rows function, which will pick out the rows of
an array with even index.
Problem 12. Please implement the mask function, which will “mask out” a value from
an array, replacing it with another value.
Problem 13. Please implement the accuracy function. This function takes a matrix of
logit scores (i.e., confidence scores) produced by a multi-class classifier along with a vector
of labels, and computes the proportion of examples that have been classified correctly.
1.4 Analysis of Matrix Multiplication (9 Points)
Python is a high-level programming language that is easy to learn and easy to read. The
beauty and simplicity of Python comes at a cost, however: Python operations are notoriously slow compared to lower-level languages such as C. Fortunately, NumPy array
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https://www.python.org/dev/peps/pep-0257/
4
operations implement many of the most common yet expensive linear algebra computations using optimized, pre-compiled C code. This means that the more you use NumPy
operations, the more efficiently your Python code will run.
In this exercise, you will compare the performance of NumPy’s matrix multiplication operator against a pure Python implementation of matrix multiplication.
Problem 14. (6 Points) Please implement the matmul pure python function, which applies matrix multiplication to its two arguments. Assume that matrices are represented as
lists of lists of numbers: each list of numbers is a row, and the full matrix is a list of rows.
You may assume that your inputs are always valid matrices.
Problem 15. (3 Points) Please run the function measure time after you have completed
Problem 14. This function will generate two random matrices, multiply them using your
pure Python implementation of matrix multiplication and using NumPy’s implementation
of matrix multiplication, and report the running time of each implementation. Please
submit the running times reported by the measure time function.
2 Word2Vec
In this part of the homework, you will apply your newfound expertise concerning Python
matrix manipulation in the world of Word2Vec vectors. Please complete the following
exercises by filling in the appropriate functions from word2vec exercise.py.
2.1 Setup
Download the file word2vec embeddings.zip from the Canvas site, which contains a set of
300-dimensional word vectors (i.e., each word is associated with a 300-dimensional vector
of real numbers). This set of word vectors comes from work by Omer Levy and Yoav
Goldberg, and it is the Bag of Words (k = 2) file listed on the following page:5
https://levyomer.wordpress.com/2014/04/25/
dependency-based-word-embeddings/
Expand the .zip archive and examine the format of the included .txt file. Note that this
file is fairly large (390MB), and will be even larger (909MB) once it is expanded, so you
should make sure that you have enough space on your computer’s disk.
In order to allow you to work with the embeddings, we have provided the load embeddings
function, which will load the word embeddings from the .txt file into Python. The function
will return a vocabulary—a list containing all the words that have embeddings—and a
matrix where each row is a word embedding. The vocabulary and the word embedding
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It will be much faster for you to download the file from Canvas than from this link.
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matrix are in the same order: the ith row of the word embedding matrix contains the
embedding for the ith list item in the vocabulary.
2.2 Exercises
Problem 16. (6 Points) Complete the definition of the function get embedding, which
looks up the embedding of a given word from the vocabulary and the word embedding
matrix.
Problem 17. (6 Points) Complete the definition of the function cosine sim which computes the cosine similarity of a pair of vectors x and y using the formula we discussed in
class:
cos(x, y) = x
>y
kxkkyk
.
Please use the function np.linalg.norm to compute kxk and kyk.
Problem 18. (6 Points) Fill in the definition of the function get neighbors of embedding.
Given a word embedding e, this function finds the top k words whose embeddings have
the highest cosine similarity with e. The top k words do not have to be in order.
Problem 19. (6 Points) Now use the functions you have defined to complete the definition
of the function get neighbors of word, which finds the top k words whose embeddings
have the greatest cosine similarity to the embedding of a given word w, not including w.
This function should operate in a manner similar to get neighbors of embedding, except
that its first argument should be a word from the vocabulary rather than a word embedding
vector. Again, the top k words to not have to been in order.
Problem 20. (3 Points) What are the 5 most similar words to the following?
• cat
• mask
• tahiti
• september
• virus
Problem 21. (6 Points) Complete the definition of the function analogy, which performs
the analogy task w1:w2::w3:x using vector arithmetic on the embeddings for the words
w1, w2 and w3:
[[x]] = [[w2]] − [[w1]] + [[w3]].
Your function should take w1, w2, and w3 as input and return the k words whose embeddings
have the greatest cosine similarity to [[w2]] − [[w1]] + [[w3]].
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Problem 22. (4 Points) Test your analogy implementation on the following examples,
looking both at the top answer and the 5 closest words:
• dog:puppy::cat:x
• france:french::spain:x
• long:longest::fat:x
• give:gives::hope:x
• dog:puppy::cow:x
• france:french::afghanistan:x
• long:longest::white:x
• give:giving::hope:x
Comment on the cases where it succeeds and those in which it does not.
3 Submission Instructions
You will submit your solutions to this assignment to CodePost, an online platform for
grading assignments. Please create an account on CodePost at the following link: https:
//codepost.io/signup/join?code=PYKUYVYOU3. You must create your account using
your @yale.edu email address and the invite code PYKUYVYOU3.
To submit your completed assignment, please upload the following files to CodePost. Please
ensure that your files have the same filenames as indicated below.
• Your edited version of numpy exercise.py, containing your solutions to Problems
9–14.
• Your edited version of word2vec exercise.py, containing your solutions to Problems
16–22.
• A Markdown document called assignment1.md, containing your responses to Problems 1–8, 15, 20, and 22.
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