Matrix factorization basics in ML Python - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to create a matrix factorization model using numpy.

ML Python

import numpy as np

# Initialize matrices
R = np.array([[5, 3, 0], [4, 0, 0], [1, 1, 0]])
num_users, num_items = R.shape
K = 2
P = np.random.rand(num_users, [1])
Q = np.random.rand(num_items, K)

Drag options to blanks, or click blank then click option'

A1

Bnum_users

Cnum_items

DK

Attempts:

3 left

💡 Hint

Common Mistakes

Using num_users or num_items as the second dimension of P instead of K.

Setting the dimension to 1 which is incorrect for latent features.

The matrix P should have dimensions (num_users x K), where K is the number of latent features.

2fill in blank

medium

Complete the code to compute the predicted rating matrix by multiplying P and Q transpose.

ML Python

predicted_ratings = np.dot(P, [1])

Drag options to blanks, or click blank then click option'

AQ

BR

CQ.T

DP.T

Attempts:

3 left

💡 Hint

Common Mistakes

Multiplying P by Q without transpose causing dimension mismatch.

Using P.T or R which are incorrect for prediction.

To get the predicted ratings matrix, multiply P (users x features) by Q transpose (features x items).

3fill in blank

hard

Fix the error in the gradient descent update step for matrix factorization.

ML Python

for i in range(num_users):
    for j in range(num_items):
        if R[i][j] > 0:
            eij = R[i][j] - np.dot(P[i, :], Q[j, :].T)
            for k in range(K):
                P[i][k] += [1] * (2 * eij * Q[j][k] - 0.01 * P[i][k])

Drag options to blanks, or click blank then click option'

A0.002

B0.02

C-0.002

D-0.02

Attempts:

3 left

💡 Hint

Common Mistakes

Using a negative learning rate causing the model to diverge.

Using too large a learning rate causing unstable updates.

The learning rate should be positive to move in the direction that reduces error.

4fill in blank

hard

Fill both blanks to complete the error calculation and regularization term in matrix factorization.

ML Python

error = 0
for i in range(num_users):
    for j in range(num_items):
        if R[i][j] > 0:
            error += (R[i][j] - np.dot(P[i, :], Q[j, :].T))[1] 2
            for k in range(K):
                error += 0.01 / 2 * (P[i][k][2] 2 + Q[j][k][2] 2)

Drag options to blanks, or click blank then click option'

A**

B*

C+

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using '*' instead of '**' which multiplies but does not square.

Using '+' or '-' which changes the meaning of the formula.

The squared error uses the power operator '**2' to square the difference and the regularization terms.

5fill in blank

hard

Fill all three blanks to create a dictionary of predicted ratings for items with predicted rating above 3.

ML Python

predicted_dict = {item: predicted_ratings[user][[1]] for user in range(num_users) for item in range(num_items) if predicted_ratings[user][item] [2] 3 and predicted_ratings[user][item] [3] 5}

Drag options to blanks, or click blank then click option'

Aitem

B>

C<

Duser

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'user' instead of 'item' as dictionary keys.

Using wrong comparison operators causing incorrect filtering.

The dictionary keys are item indices, and the condition filters ratings greater than 3 and less than 5.

Practice

1. What is the main goal of matrix factorization in machine learning?

easy

A. To break a large matrix into smaller matrices to find hidden patterns

B. To increase the size of a matrix for better accuracy

C. To convert a matrix into a single number

D. To sort the matrix rows in ascending order

Solution

1. Step 1: Understand matrix factorization purpose

   Matrix factorization breaks a big matrix into smaller ones to reveal hidden structures or patterns.

2. Step 2: Compare options with this purpose

   Only To break a large matrix into smaller matrices to find hidden patterns correctly describes this goal; others describe unrelated tasks.

3. Final Answer:

   To break a large matrix into smaller matrices to find hidden patterns -> Option A

4. Quick Check:

   Matrix factorization = breaking matrix into smaller ones [OK]

Hint: Matrix factorization means splitting matrix to find patterns [OK]

Common Mistakes:

• Thinking matrix factorization increases matrix size
• Confusing factorization with sorting or summarizing
• Assuming it converts matrix to a single value

2. Which of the following is the correct way to represent matrix factorization of a matrix R into two matrices P and Q?

easy

A. R = P + Q

B. R = P * Q (matrix multiplication)

C. R = P - Q

D. R = P / Q

Solution

1. Step 1: Recall matrix factorization formula

   Matrix factorization expresses R as the product of two smaller matrices P and Q.

2. Step 2: Identify correct mathematical operation

   Matrix multiplication (*) is used, not addition, subtraction, or division.

3. Final Answer:

   R = P * Q (matrix multiplication) -> Option B

4. Quick Check:

   Matrix factorization = matrix multiplication [OK]

Hint: Matrix factorization uses multiplication, not addition [OK]

Common Mistakes:

• Using addition or subtraction instead of multiplication
• Confusing element-wise operations with matrix multiplication
• Thinking division applies to matrices here

3. Given the following Python code using NumPy for matrix factorization approximation:

import numpy as np
R = np.array([[5, 3], [4, 0]])
P = np.array([[1, 0], [0, 1]])
Q = np.array([[5, 3], [4, 2]])
approx = np.dot(P, Q)
print(approx)

What is the output printed?

medium

A. [[5 3]
[4 2]]

B. [[5 3]
[0 0]]

C. [[1 0]
[0 1]]

D. [[9 3]
[4 2]]

Solution

1. Step 1: Calculate matrix multiplication np.dot(P, Q)

   Multiplying P=[[1,0],[0,1]] by Q=[[5,3],[4,2]] results in Q itself because P is identity-like.

2. Step 2: Confirm output matches Q

   The result is [[5,3],[4,2]], matching [[5 3] [4 2]] exactly.

3. Final Answer:

   [[5 3] [4 2]] -> Option A

4. Quick Check:

   Identity matrix times Q = Q [OK]

Hint: Identity matrix times any matrix equals that matrix [OK]

Common Mistakes:

• Confusing element-wise multiplication with dot product
• Misreading P as zero matrix
• Expecting zeros in output due to zero in R

4. Consider this Python snippet for matrix factorization update step:

import numpy as np
R = np.array([[5, 3], [4, 0]])
P = np.array([[1, 0], [0, 1]])
Q = np.array([[5, 3], [4, 2]])
for i in range(2):
    for j in range(2):
        if R[i][j] != 0:
            error = R[i][j] - np.dot(P[i, :], Q[:, j])
            P[i, :] += 0.01 * error * Q[:, j]
print(P)

What is the main issue with this code?

medium

A. The update step incorrectly updates P using Q's columns instead of rows

B. The error calculation uses wrong indices for R

C. The learning rate 0.01 is too large causing divergence

D. The code misses updating Q matrix

Solution

1. Step 1: Analyze update step for matrix factorization

   Matrix factorization updates both P and Q to reduce error, but code updates only P.

2. Step 2: Identify missing update for Q

   Q is never updated, so factorization won't improve fully.

3. Final Answer:

   The code misses updating Q matrix -> Option D

4. Quick Check:

   Both P and Q must be updated [OK]

Hint: Update both P and Q matrices in factorization [OK]

Common Mistakes:

• Thinking only P needs updating
• Confusing error calculation indices
• Assuming learning rate causes error here

5. You have a user-item rating matrix with many missing values. Which approach best uses matrix factorization to predict missing ratings?

hard

A. Fill missing values with zeros and factorize the full matrix directly

B. Ignore missing values and factorize only the submatrix of known ratings

C. Factorize the matrix into two smaller matrices and use gradient descent to minimize error only on known ratings

D. Use matrix addition to combine user and item matrices

Solution

1. Step 1: Understand handling missing values in matrix factorization

   Missing values should not be treated as zeros; only known ratings guide factorization.

2. Step 2: Use gradient descent minimizing error on known ratings

   This approach updates factor matrices to predict missing values accurately.

3. Final Answer:

   Factorize the matrix into two smaller matrices and use gradient descent to minimize error only on known ratings -> Option C

4. Quick Check:

   Minimize error on known ratings for prediction [OK]

Hint: Train only on known ratings, predict missing [OK]

Common Mistakes:

• Filling missing values with zeros distorts learning
• Ignoring missing values loses data context
• Using addition instead of multiplication for factorization