Cosine Similarity

Cosine similarity measures the angle between two vectors and ignores their length. It is the default way to compare embeddings, because for text the direction carries the meaning and the magnitude mostly carries frequency.

cos(a, b) = (a . b) / (||a|| * ||b||)

The result is in [-1, 1]: 1 for vectors pointing the same way, 0 for perpendicular, -1 for opposite.

If either vector has zero length the angle is undefined, so return 0.0.

Function signature

def cosine_similarity(a, b):

Example

Input

cosine_similarity(np.array([1.0, 0.0]), np.array([1.0, 1.0]))

Output

0.7071067811865475

Reasoning

The dot product is 1, and the norms are 1 and sqrt(2). So the cosine is 1 / sqrt(2), about 0.7071, which is a 45 degree angle.

Constraints and notes

• Return 0.0 when either vector has zero length.
• The result is in [-1, 1].