Cosine similarity in NLP - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the dot product of two vectors.

NLP

dot_product = sum(a * b for a, b in zip(vec1, [1]))

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

Avec3

Bvec2

Cvec1

Dvector

Attempts:

3 left

💡 Hint

Common Mistakes

Using the same vector twice, which calculates the dot product of a vector with itself.

The dot product multiplies corresponding elements of two vectors and sums them. Here, vec2 is the second vector to multiply with vec1.

2fill in blank

medium

Complete the code to calculate the magnitude (length) of a vector.

NLP

magnitude = sum(x[1]2 for x in vec) ** 0.5

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

A**

B+

C*

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using addition or multiplication instead of exponentiation to square elements.

The magnitude is the square root of the sum of squares of vector elements. So each element is squared using ** 2.

3fill in blank

hard

Fix the error in the cosine similarity formula by completing the denominator.

NLP

cos_sim = dot_product / (magnitude1 [1] magnitude2)

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

A*

B/

C-

D+

Attempts:

3 left

💡 Hint

Common Mistakes

Using addition or division instead of multiplication in the denominator.

The denominator of cosine similarity is the product of the magnitudes of the two vectors, so the operator is *.

4fill in blank

hard

Fill both blanks to complete the dictionary comprehension that maps words to their vector lengths if length is greater than 3.

NLP

lengths = {word: [1] for word in words if len(word) [2] 3}

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

Alen(word)

B>

C<

Dword

Attempts:

3 left

💡 Hint

Common Mistakes

Using the word itself as value instead of its length.

Using < instead of > in the condition.

The dictionary comprehension creates a mapping from each word to its length, but only for words longer than 3 characters. So len(word) is the value, and the condition uses >.

5fill in blank

hard

Fill all three blanks to create a dictionary of words mapped to their lengths, but only include words with length greater than 2.

NLP

result = [1]: [2] for [3] in word_list if len([3]) > 2}

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

Aword

Blen(word)

Cword_list

Attempts:

3 left

💡 Hint

Common Mistakes

Using the list name as loop variable.

Mixing variable names inconsistently.

The dictionary comprehension maps each word to its length len(word). The loop variable is word iterating over word_list. The condition filters words longer than 2.

Practice

1. What does cosine similarity measure between two vectors?

easy

A. The difference in vector lengths

B. How close the vectors point in the same direction

C. The sum of vector elements

D. The distance between vector origins

Solution

1. Step 1: Understand vector comparison

   Cosine similarity compares the angle between two vectors, not their length or sum.

2. Step 2: Interpret cosine similarity meaning

   A value close to 1 means vectors point in the same direction, showing similarity.

3. Final Answer:

   How close the vectors point in the same direction -> Option B

4. Quick Check:

   Cosine similarity = direction closeness [OK]

Hint: Cosine similarity checks angle, not length or sum [OK]

Common Mistakes:

• Confusing cosine similarity with Euclidean distance
• Thinking it measures vector length difference
• Assuming it sums vector values

2. Which of the following is the correct formula for cosine similarity between vectors A and B?

easy

A. \( \frac{\|A\|}{\|B\|} \)

B. \( \|A - B\| \)

C. \( \frac{A \cdot B}{\|A\| \times \|B\|} \)

D. \( A + B \)

Solution

1. Step 1: Recall cosine similarity formula

   Cosine similarity is the dot product of vectors divided by the product of their lengths.

2. Step 2: Match formula to options

   \( \frac{A \cdot B}{\|A\| \times \|B\|} \) matches the formula \( \frac{A \cdot B}{\|A\| \times \|B\|} \), others do not.

3. Final Answer:

   \( \frac{A \cdot B}{\|A\| \times \|B\|} \) -> Option C

4. Quick Check:

   Cosine similarity = dot product / product of norms [OK]

Hint: Look for dot product over product of lengths [OK]

Common Mistakes:

• Choosing Euclidean distance formula
• Adding vectors instead of dot product
• Dividing norms instead of multiplying

3. Given vectors A = [1, 2, 3] and B = [4, 5, 6], what is the cosine similarity (rounded to 2 decimals)?

medium

A. 0.97

B. 0.83

C. 0.74

D. 0.56

Solution

1. Step 1: Calculate dot product of A and B

   Dot product = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32

2. Step 2: Calculate norms of A and B

   Norm A = sqrt(1^2 + 2^2 + 3^2) = sqrt(14) ≈ 3.74; Norm B = sqrt(4^2 + 5^2 + 6^2) = sqrt(77) ≈ 8.77

3. Step 3: Compute cosine similarity

   Cosine similarity = 32 / (3.74 * 8.77) ≈ 32 / 32.83 ≈ 0.9749 rounded to 0.97

4. Step 4: Check closest option

   0.97 matches the value rounded to 2 decimals.

5. Final Answer:

   0.97 -> Option A

6. Quick Check:

   Dot product / (norms product) ≈ 0.97 [OK]

Hint: Calculate dot product and divide by product of lengths [OK]

Common Mistakes:

• Forgetting to take vector norms
• Mixing up dot product with element-wise multiplication
• Rounding too early causing wrong answer

4. What is wrong with this Python code to compute cosine similarity?

import numpy as np

def cosine_sim(a, b):
    return np.dot(a, b) / np.linalg.norm(a + b)

A = np.array([1, 0])
B = np.array([0, 1])
print(cosine_sim(A, B))

medium

A. It should add vectors before dot product

B. It uses np.dot instead of np.cross

C. It misses normalizing vectors before dot product

D. It divides by norm of sum instead of product of norms

Solution

1. Step 1: Analyze denominator in code

   The code divides by norm of (a + b), but cosine similarity requires product of norms of a and b.

2. Step 2: Understand correct formula

   Correct denominator is np.linalg.norm(a) * np.linalg.norm(b), not norm of sum.

3. Final Answer:

   It divides by norm of sum instead of product of norms -> Option D

4. Quick Check:

   Denominator must be product of norms [OK]

Hint: Denominator is product of norms, not norm of sum [OK]

Common Mistakes:

• Using norm of sum instead of product
• Confusing dot product with cross product
• Normalizing vectors before dot product unnecessarily

5. You have two text documents represented as TF-IDF vectors: doc1 = [0, 1, 2, 0] and doc2 = [1, 0, 1, 1]. Which step is best to improve cosine similarity comparison for very sparse vectors?

hard

A. Normalize vectors to unit length before computing cosine similarity

B. Add the vectors element-wise before similarity

C. Use Euclidean distance instead of cosine similarity

D. Ignore zero elements in vectors

Solution

1. Step 1: Understand sparse vector challenges

   Sparse vectors have many zeros; normalizing to unit length ensures fair angle comparison.

2. Step 2: Identify best practice for cosine similarity

   Normalizing vectors before cosine similarity avoids bias from vector length differences.

3. Final Answer:

   Normalize vectors to unit length before computing cosine similarity -> Option A

4. Quick Check:

   Normalization improves cosine similarity on sparse data [OK]

Hint: Always normalize vectors before cosine similarity [OK]

Common Mistakes:

• Adding vectors instead of comparing
• Switching to Euclidean distance without reason
• Ignoring zeros instead of normalizing