Embeddings and Vector Search

Encode text, pick a metric, size the vectors, choose an index and fix bad recall. Python, NumPy and FAISS.

Retrieve, then rerank#

encoder = SentenceTransformer("BAAI/bge-small-en-v1.5")384 dim

embeddings = encoder.encode(corpus, normalize_embeddings=True)unit vectors

q_emb = encoder.encode([query], normalize_embeddings=True)

sims = (embeddings @ q_emb.T).flatten()dot equals cosine

order = np.argsort(-sims)[:top_k]best first

def truncate(vectors, dim):Matryoshka cut

out = vectors[:, :dim]

return out / np.linalg.norm(out, axis=1, keepdims=True)renormalize

emb_256 = truncate(emb, 256)

index = faiss.IndexFlatL2(dimension)exact search

index.add(embeddings)

distances, indices = index.search(query_vectors, k=5)

Task	Code
Truncate one vector	`truncate_matryoshka(vec, 64)`
Binarize a vector	`binarize(vec)`
Search by Hamming distance	`engine.index.search(query_vec, top_k=5, metric="hamming")`
Fuse dense and sparse ranks	`rrf_fuse([dense_ranked[:5], sparse_scores[:5]])`
Load BGE-M3	`BGEM3FlagModel("BAAI/bge-m3", use_fp16=True)`
Run an analogy	`analogy(vocab, W, "man", "king", "woman")`
Vector for an unseen word	`fasttext_vector(word, ngram_table)`
Evaluate on MTEB tasks	`evaluation.run(encoder, output_folder="./mteb-results")`

Command	What it does
`IndexFlatIP`	Exact inner product. Cosine on unit vectors. About 100k vectors
`IndexFlatL2`	Exact L2, no training. Up to about 1M vectors
`IndexIVFFlat`	K cells, searches the closest few. Needs training. About 10M
`IndexIVFPQ`	IVF plus product quantization. 100M and more
`IndexHNSW`	Graph index. Fastest for many queries. Large index
LSH	Random hash projections. Near-duplicate detection
KD-trees, ball trees	Exact. Low and medium dimension only

Symptom	Cause	Fix
bge recall 3 to 5 points low	No query prefix	Prepend the prefix from the model card
Truncated vectors rank badly	No renormalize after the cut	Divide by the norm again
Recall falls after truncation	Cut too far, or no Matryoshka training	Validate on your eval set
Nearest neighbors change	Cosine and L2 conventions mixed	Match the training metric
Query and doc do not match	Asymmetric model	Check the model card
`cosine: dim mismatch`	Vectors of different size	One model and one dim for both
Long docs match nothing	One vector averages all topics	Chunk 256 to 512 tokens, 50 overlap
Unseen word has no vector	Word2Vec and GloVe have no fallback	Use FastText n-grams

Cosine or dot product?: On L2-normalized vectors the choice does not matter.
Why ANN?: Brute force at 1M by 1536 is 1.5 billion multiply-adds per query.
Why rerank?: A cross-encoder reads query and doc together. It is slower but more accurate.
Why RRF, not a score sum?: BM25 is unbounded. Cosine is in -1 to 1. Ranks compare.
Which number for a RAG retriever?: Recall@k. No right passage, no answer.