Vector Quantization

Why Quantize

A float32 1024-dim embedding is 4 KB. 100M vectors = 400 GB. Quantization cuts this 4-128x with small recall cost — especially when paired with rescoring.

Scalar Quantization (int8 / int4)

Linear mapping: each float32 dim scaled to an 8-bit or 4-bit integer with per-index or per-dim scale/zero-point.

import numpy as np

def int8_quantize(x: np.ndarray):
    # Symmetric per-dim scale
    scale = np.max(np.abs(x), axis=0) / 127
    q = np.round(x / scale).clip(-128, 127).astype(np.int8)
    return q, scale

def int8_dot(q: np.ndarray, q_q: np.ndarray, s_a: np.ndarray, s_b: np.ndarray) -> np.ndarray:
    return (q.astype(np.int32) @ q_q.astype(np.int32).T) * (s_a * s_b).sum()

For production, use framework support (Faiss, pgvector halfvec/int8, Qdrant scalar quantizer, Elasticsearch int8_hnsw) — they handle numerical issues and SIMD.

Faiss scalar quantization

import faiss

quantizer = faiss.IndexFlatL2(d)
index = faiss.IndexIVFScalarQuantizer(
    quantizer, d, nlist=1024,
    qtype=faiss.ScalarQuantizer.QT_8bit, metric=faiss.METRIC_L2,
)
index.train(sample)
index.add(corpus)
index.nprobe = 32

Binary Quantization

Each float becomes 1 bit: bit = (value > 0). Distance becomes Hamming distance (XOR + popcount), which is 10-40x faster than cosine on modern CPUs.

import numpy as np

def binary_quantize(vecs: np.ndarray) -> np.ndarray:
    return np.packbits((vecs > 0).astype(np.uint8), axis=1)   # (N, d/8)

def hamming_topk(qbits: np.ndarray, cbits: np.ndarray, k: int):
    # XOR + popcount via numpy
    xor = np.bitwise_xor(cbits, qbits)
    dists = np.unpackbits(xor, axis=1).sum(axis=1)
    topk = np.argpartition(dists, k)[:k]
    return topk[np.argsort(dists[topk])]

Recovery via Rescoring

Binary alone loses 5-15% recall. Fix: oversample on binary, then rescore the shortlist with full float vectors.

def search_binary_then_rescore(qvec: np.ndarray, corpus: np.ndarray,
                               cbits: np.ndarray, top_k: int, oversample: int = 20):
    qbits = binary_quantize(qvec[None])[0]
    shortlist = hamming_topk(qbits, cbits, top_k * oversample)
    # Rescore with full float32
    sub = corpus[shortlist]
    scores = sub @ qvec
    order = np.argsort(-scores)[:top_k]
    return shortlist[order], scores[order]

Typical result: ~99% of full-float recall at 5-10x lower search latency and 32x less stored vector data.

Binary in pgvector

-- Column
embedding_bit bit(1024)

-- Populate
UPDATE chunks SET embedding_bit = binary_quantize(embedding::vector)::bit(1024);

-- Index (pgvector 0.7+)
CREATE INDEX ON chunks USING hnsw (embedding_bit bit_hamming_ops);

-- Search then rescore
WITH shortlist AS (
  SELECT id, embedding
  FROM chunks
  ORDER BY embedding_bit <~> binary_quantize($1::vector)::bit(1024)
  LIMIT 100
)
SELECT id, 1 - (embedding <=> $1::vector) AS score
FROM shortlist
ORDER BY embedding <=> $1::vector
LIMIT 10;

Binary in Qdrant

from qdrant_client import models

client.update_collection(
    "docs",
    quantization_config=models.BinaryQuantization(
        binary=models.BinaryQuantizationConfig(always_ram=True),
    ),
)
client.query_points(
    "docs",
    query=qvec.tolist(),
    using="dense",
    limit=10,
    search_params=models.SearchParams(
        quantization=models.QuantizationSearchParams(
            rescore=True, oversampling=3.0,
        ),
    ),
)

Product Quantization (PQ)

Split each d-dim vector into m subvectors of size d/m. Quantize each subvector to one of ksub = 2^nbits centroids learned by k-means on a training sample. Stored code: m * nbits bits per vector.

Faiss PQ

import faiss

m, nbits = 64, 8                             # 64 bytes per vector
index = faiss.IndexPQ(d, m, nbits)
index.train(sample)                          # needs ~256 * 2^nbits points
index.add(corpus)
D, I = index.search(queries, 10)

IVF + PQ (standard huge-scale combo)

quantizer = faiss.IndexFlatL2(d)
index = faiss.IndexIVFPQ(quantizer, d, nlist=8192, m=64, nbits=8)
index.train(sample)
index.add(corpus)
index.nprobe = 64

Faiss computes query-to-centroid distances once per query (m * 256 values) and sums them per candidate, which is very SIMD-friendly.

Optimized PQ (OPQ)

Apply a learned orthogonal rotation before PQ so subvectors are better balanced. Gains ~1-3% recall at the same code size.

import faiss
opq = faiss.OPQMatrix(d, m)
opq.train(sample)
rotated = opq.apply_py(corpus)
pq = faiss.IndexPQ(d, m, nbits)
pq.train(rotated)
pq.add(rotated)
# Or use the pre-built combo:
index = faiss.index_factory(d, f"OPQ{m}_64,IVF8192,PQ{m}x{nbits}")

Residual Quantization (RQ)

Quantize the residual (original - first quantized approximation) with another codebook. Stack multiple stages for fine-grained quantization.

Use cases: where PQ at a given rate underfits — RQ gets closer to float32 at the same bit budget. Implemented in Faiss as IndexResidualQuantizer, and underlies ScaNN's anisotropic quantizer.

index = faiss.IndexResidualQuantizer(d, nsplits=4, nbits=8)
index.train(sample)
index.add(corpus)

Rescoring Workflow (Universal Pattern)

1. Store FULL precision vectors in cold storage (or in the same DB as a second column).
2. Index QUANTIZED vectors for ANN search.
3. Search quantized, retrieve `oversample * k` candidates.
4. Rescore candidates with full vectors (dot product / cosine).
5. Return true top-k.

This pattern recovers 99%+ of full-precision recall for binary/PQ and costs only the oversample factor in extra candidates.

def hybrid_search(qvec, k=10, oversample=20):
    candidates = quantized_index.search(qvec, k * oversample)  # fast
    full_vecs = full_store.get_many([c.id for c in candidates])
    scored = sorted(
        zip(candidates, full_vecs),
        key=lambda x: -np.dot(qvec, x[1]),
    )
    return scored[:k]

Storage Math Reference

For N vectors of dimension d:

Example: 100M x 1024-d

• float32: 400 GB
• float16: 200 GB
• int8: 102 GB
• binary: 12.5 GB
• PQ(64, 8): 6.4 GB
• IVFPQ(64, 8, 8192): 6.4 GB + 32 MB centroids

Accuracy vs Memory Curve (rough)

• int8: keep 99%+ of float32 recall. Always safe default.
• PQ(d/16, 8): keep 95-98% with rescoring. Great for 10M+.
• Binary: keep 95-99% WITH rescoring, 85-92% without.
• PQ(d/64, 8): keep 85-92%, even with rescoring; use only at extreme scale.

Always measure on YOUR data. Quantization quality depends on embedding distribution.

Anti-Patterns

Production Checklist

• Recall@k measured against brute-force ground truth at chosen quantization
• Oversampling factor tuned (typically 2-10x)
• Rescoring pipeline implemented (full vectors retrievable by ID)
• Training sample size meets quantizer requirements
• Retraining schedule for IVF/PQ centroids documented
• Memory budget verified post-quantization
• Metric consistency (cosine vs L2) across quantized and full paths
• Fallback to float32 for critical queries if needed