Cosine similarity and text retrieval

The problem

Text retrieval needs a score between a query vector and each document vector. A dot product rewards shared coordinates, but its magnitude also grows with vector length. Long documents can rank highly simply because they contain more weighted terms.

Cosine similarity compares direction. It divides the dot product by both vector lengths, producing a score from -1 to 1 for general real vectors. Nonnegative TF-IDF vectors produce scores from 0 to 1.

Work through a small example

Let query q=[1,1]. Document a=[2,2] points in exactly the same direction, so cosine is 1. Document b=[2,0] shares one feature, giving 2/(sqrt(2)*2)=0.707.

Scaling does not change direction: cos(q,100q)=1. This is useful when repetition or document length should matter less than the relative feature pattern.

The general rule

cos(q,d)=(q dot d)/(||q||_2 ||d||_2). The L2 norm is the square root of summed squared coordinates. If either norm is zero, the expression is undefined; retrieval code should return no evidence or a documented zero, not divide silently.

For a query matrix shaped (Q,D) and normalized document matrix shaped (N,D), matrix multiplication Q @ documents.T returns all scores with shape (Q,N).

Implement and inspect

import numpy as np

query = np.array([1.0, 1.0])
documents = np.array([[2.0, 2.0], [2.0, 0.0]])

query_norm = np.linalg.norm(query)
document_norms = np.linalg.norm(documents, axis=1, keepdims=True)
if query_norm == 0 or np.any(document_norms == 0):
    raise ValueError("cosine similarity needs nonzero vectors")
query = query / query_norm
documents = documents / document_norms
scores = documents @ query
print(np.round(scores, 3).tolist())
print(np.argsort(-scores).tolist())

Expected output is similarities [1.0,0.707] and rank order [0,1].

documents has shape (2,2); query has shape (2,); scores have shape (2,). Negating scores makes ascending argsort return highest similarity first.

Engineering checks

• Normalize along the feature axis, not across documents.
• Guard zero vectors caused by empty or all-out-of-vocabulary (OOV) text, meaning words absent from the fitted vocabulary.
• Exclude the query document when evaluating nearest neighbours.
• Use a stable tie rule so equal scores are reproducible.

For sparse TF-IDF, compute dot products only on shared nonzero coordinates. Dense conversion can exhaust memory without changing the mathematics.

Go deeper

Cosine ignores norm by design. That can remove nuisance document length, but vector norm may carry confidence or frequency information in learned representations. Compare cosine with dot product using the downstream metric.

Nearest-neighbour quality also depends on the representation. Cosine cannot recover word order discarded by bag of words. In high dimensions, similarity distributions can concentrate, and approximate indexes trade exact ranking for speed.

Practice and recap

Question: What is cosine between [1,0] and [-3,0]?

Cosine ranks directions rather than magnitudes. Normalize consistently, define zero-vector behavior, and evaluate the full representation-and-retrieval pipeline.