Cosine similarity top-k search

Problem

A vector search finds the documents whose embeddings point in the same direction as a query embedding. Cosine similarity measures that direction and ignores the length of the vectors. The retrieval step of a RAG system starts with this search.

Task

Write three functions.

1. normalize_rows(X) returns a copy of X with each row divided by its L2 norm.
2. cosine_similarity_matrix(Q, D) returns the cosine similarity of each query row against each document row.
3. top_k(Q, D, k) returns, for each query, the indices of the k documents with the highest cosine similarity.

The cosine similarity of vectors q and d is \frac{q \cdot d}{\lVert q \rVert \, \lVert d \rVert}.

Zero vector rule: a row with norm 0 has no direction. normalize_rows leaves it as a row of zeros. Its cosine similarity with any vector is 0.0.

Tie rule: top_k sorts the scores from high to low. When two documents have the same score, the document with the lower index comes first.

Input and output

• normalize_rows returns a new float array of shape (n, dim).
• cosine_similarity_matrix returns a float array of shape (n_queries, n_docs). Entry (i, j) is the score of query i against document j.
• top_k returns an int array of shape (n_queries, k). Row i holds document indices, best first.

Constraints

• 1 <= n_queries <= 64, 1 <= n_docs <= 1000 and 1 <= dim <= 1024.
• 1 <= k <= n_docs.
• Rows of Q and D can be zero vectors.
• Do not change the input arrays.
• Use numpy only.

Examples

pythonCopynormalize_rows(np.array([[3.0, 4.0], [0.0, 0.0], [0.0, -2.0]]))

textCopy[[ 0.6  0.8]
 [ 0.   0. ]
 [ 0.  -1. ]]

pythonCopyQ = np.array([[1.0, 0.0], [1.0, 1.0]])
D = np.array([[2.0, 0.0], [0.0, 3.0], [1.0, 1.0], [-1.0, 0.0]])
cosine_similarity_matrix(Q, D)

textCopy[[ 1.          0.          0.70710678 -1.        ]
 [ 0.70710678  0.70710678  1.         -0.70710678]]

pythonCopytop_k(Q, D, 2)

textCopy[[0 2]
 [2 0]]

For the second query, documents 0 and 1 have the same score. The tie rule puts document 0 first.