Cosine Similarity is a metric used to measure how similar two vectors are, regardless of their magnitude. It is frequently used in text analysis, recommendation systems, and clustering tasks, where the orientation of data (rather than its scale) is more important.
The Cosine Similarity between two non-zero vectors A and B is defined as:
Where :
- A ⋅ B is the dot product of vectors A and B.
- ||A|| is the Euclidean (L2) norm of A.
- ||B|| is the Euclidean (L2) norm of B.
Why Cosine Similarity?
Unlike Euclidean distance, which measures the magnitude of difference between two points, cosine similarity focuses on the direction of vectors. This makes it particularly useful for comparing high-dimensional data like word embeddings or TF-IDF scores in natural language processing.
Let’s now look at some examples.
Example 1: Cosine Similarity Between Two 1D Vectors
In the example below we compute the cosine similarity between the two vectors (1-d NumPy arrays). To define a vector here we can also use the Python Lists.
import numpy as np
from numpy.linalg import norm
A = np.array([2, 1, 2, 3, 2, 9])
B = np.array([3, 4, 2, 4, 5, 5])
# compute cosine similarity
cosine = np.dot(A, B) / (norm(A) * norm(B))
print("Cosine Similarity:", cosine)
Output:
Cosine Similarity: 0.8188504723485274Explanation:
- np.dot(A, B) computes the dot product.
- norm(A) and norm(B) compute the magnitude (Euclidean norm).
- The result is a scalar value between -1 and 1 representing similarity.
Example 2: Cosine Similarity Between a Vector and a Batch of Vectors
In the example below we compute the cosine similarity between the two vectors (1-d NumPy arrays).
Python
import numpy as np
from numpy.linalg import norm
A = np.array([[2, 1, 2], [3, 2, 9], [-1, 2, -3]])
B = np.array([3, 4, 2])
cosine = np.dot(A, B) / (norm(A, axis=1) * norm(B))
print("Cosine Similarity:", cosine)
Output:
Cosine Similarity: [ 0.86657824 0.67035541 -0.04962917]Explanation:
- A has 3 vectors (rows), and B is a single vector.
- Cosine similarity is computed row-wise.
- Negative similarity indicates opposite direction.
Example 3: Cosine Similarity Between Two Matrices (Row-wise)
We now compute similarity between corresponding rows in two 2D arrays.
Python
import numpy as np
from numpy.linalg import norm
A = np.array([[1, 2, 2], [3, 2, 2], [-2, 1, -3]])