hadamax 0.1.0

Fast compressed ANN search via randomized Hadamard transform and optimal Gaussian scalar quantization. Pure NumPy, no heavy dependencies.

hadamax

Fast compressed approximate nearest-neighbor search. Pure NumPy. No heavy dependencies.

hadamax implements the TurboQuant compression pipeline — randomized Hadamard transform followed by optimal Gaussian scalar quantization (Lloyd-Max) — as a self-contained Python library for embedding vector search. It achieves 8–12× compression with >0.92 recall@10 against float32 brute-force, using only NumPy.

pip install hadamax

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Quick start

import numpy as np
from hadamax import HadaMaxIndex

# Build index
idx = HadaMaxIndex(dim=384, bits=4)          # 4-bit, ~8x compression
idx.add_batch(ids=list(range(N)), vectors=embeddings)

# Query
results = idx.search(query_vector, k=10)     # [(id, score), ...]

# Persist
idx.save("my_index.hdmx")
idx2 = HadaMaxIndex.load("my_index.hdmx")   # atomic save, v1/v2 compatible

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Technical background

The problem: embedding vectors are expensive

Modern embedding models produce float32 vectors of dimension $d \in {384, 768, 1536}$. Storing $N$ vectors requires $4Nd$ bytes and brute-force search costs $O(Nd)$ per query. For $N = 1\text{M}$, $d = 384$: 1.5 GB RAM, and inner products dominate inference time.

Product Quantization (PQ) splits vectors into $M$ sub-vectors and quantizes each independently. It is effective but requires training a $K$-means codebook, which is expensive and data-dependent. Random Binary Quantization (RaBitQ, 1-bit) is fast but coarse.

TurboQuant (Zandieh et al., ICLR 2026, arXiv:2504.19874) achieves near-optimal distortion at $b$ bits per coordinate without training codebooks, by first rotating the space with a randomized Hadamard transform to make coordinates approximately Gaussian, then quantizing each coordinate independently with the optimal scalar quantizer for $\mathcal{N}(0,1)$.

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Algorithm

Step 1 — Normalize

Given a raw embedding $v \in \mathbb{R}^d$, compute the unit vector $\hat{v} = v / |v|$ and store $|v|$ separately (float32, 4 bytes).

Step 2 — Randomized Hadamard Transform (RHT)

Pad $\hat{v}$ to the next power of 2 ($d' = 2^{\lceil \log_2 d \rceil}$), then apply:

$$x = \frac{1}{\sqrt{d'}} , H , D , \hat{v}$$

where $D = \text{diag}(\sigma_1, \ldots, \sigma_{d'})$ with $\sigma_i \stackrel{\text{i.i.d.}}{\sim} \pm 1$ (deterministic from a seed), and $H$ is the unnormalized Walsh-Hadamard matrix.

By the Johnson-Lindenstrauss lemma, each coordinate $x_i \approx \mathcal{N}(0, 1/d')$. After rescaling $\tilde{x} = x \cdot \sqrt{d'}$, the coordinates are approximately $\mathcal{N}(0, 1)$ regardless of the original distribution of $v$.

Complexity: $O(d \log d)$ — no matrix multiplication.

Step 3 — Lloyd-Max scalar quantization

The optimal scalar quantizer for $\mathcal{N}(0,1)$ at $b$ bits partitions $\mathbb{R}$ into $2^b$ intervals and assigns each the conditional mean as reconstruction value. These boundaries and centroids are precomputed and hardcoded in hadamax._codebooks:

bits	levels	MSE distortion	bytes/coord
2	4	0.1175	0.25
3	8	0.0311	0.375
4	16	0.0077	0.5

The quantized vector is stored as a uint8 index matrix, bit-packed to $b/8$ bytes per coordinate.

Step 4 — Approximate inner product

At search time, the query $q$ is rotated (not quantized) and the approximate cosine similarity is computed as:

$$\hat{\langle \hat{q}, \hat{v} \rangle} = \frac{1}{d'} \sum_{i=1}^{d'} c_{\tilde{q}i} \cdot c{\hat{v}_i}$$

where $c_j$ is the centroid for index $j$. This is a single float16 matrix-vector product.

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TurboQuant_prod: unbiased estimator with QJL correction

The MSE quantizer introduces a systematic downward bias in the inner product estimate. The use_prod=True mode corrects this using a Quantized Johnson-Lindenstrauss (QJL) residual:

1. Quantize at $(b-1)$ bits MSE, compute residual $r = \tilde{x} - \hat{x}_{\text{MSE}}$
2. Store $\text{sign}(S r)$ as a 1-bit vector (int8 in practice), where $S \in \mathbb{R}^{d' \times d'}$ is a fixed random Gaussian matrix
3. Store $|r| / \sqrt{d'}$ (one float32 per vector)

At query time, the correction term is:

$$\text{correction}_i = \frac{\sqrt{\pi/2}}{d'} \cdot |r_i| \cdot \bigl\langle S \hat{q}, \text{sign}(S r_i) \bigr\rangle$$

This follows from Lemma 4 of Zandieh et al. (2025): $\mathbb{E}[\text{sign}(S r) \mid r] = \sqrt{2/\pi} \cdot Sr / |Sr|$, which gives an unbiased estimate of $\langle r, \hat{q} \rangle$ up to a correction involving $|r|$.

When to use use_prod:

• When you need accurate inner product magnitudes (e.g., KV-cache compression, attention approximation)
• Not recommended for pure ranking/NNS — the added QJL variance degrades recall@k relative to MSE-only at equal total bits

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Compression ratios

For $N$ vectors of dimension $d = 384$:

Backend	Bytes/vector	Ratio vs float32
float32	1536	1.0×
4-bit hadamax	192 + 4	7.9×
3-bit hadamax	144 + 4	10.4×
2-bit hadamax	96 + 4	15.4×
int8 (naïve)	384 + 4	3.9×

The 4-byte overhead is the per-vector norm. The float16 search cache (centroid expansions) is allocated lazily and evicted on writes.

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Recall benchmarks

Measured on synthetic unit-sphere vectors ($d=384$, $N=10{,}000$, 100 queries). Baseline: exact cosine float32 brute-force.

bits	recall@1	recall@10	recall@50
2	0.72	0.83	0.91
3	0.81	0.91	0.96
4	0.86	0.93	0.95

Recall improves with clustered (real-world) data. On BGE-small-en embeddings from mixed document corpora, 4-bit achieves recall@10 ≈ 0.95.

Note on published results: The TurboQuant paper (Zandieh et al., 2025) reports recall up to 0.99, measured against HNSW graph navigation (not brute-force float32), on GloVe $d=200$ data, using recall@1 with large $k_{\text{probe}}$. These conditions differ from the above; both results are correct under their respective definitions.

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File format

Files use the .hdmx extension.

magic:   4 bytes  "HDMX"
version: 4 bytes  uint32 (1 or 2)
dim:     4 bytes  uint32  — original embedding dimension
bits:    4 bytes  uint32  — total bits (2, 3, or 4)
seed:    4 bytes  uint32  — rotation seed
n:       4 bytes  uint32  — number of vectors
flags:   4 bytes  uint32  — bit-0: use_prod  [v2 only]
─────────────────────────────────────────
packed_len: 4 bytes  uint32
indices:    packed_len bytes  — bit-packed uint8 MSE indices
norms:      n × 4 bytes  float32  — per-vector original norms
[prod only]
  qjl_signs:  n × pdim bytes  int8  — sign(S·r) per vector
  rnorms:      n × 4 bytes  float32  — ‖r‖/√d per vector
ids:    n × (2 + len) bytes  — uint16 length-prefixed UTF-8 strings

Saves are atomic on POSIX: writes to .hdmx.tmp then os.replace().

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API reference

HadaMaxIndex(dim, bits=4, seed=0, use_prod=False)

Parameter	Type	Default	Description
dim	int	—	Embedding dimension
bits	int	4	Bits per coordinate (2, 3, or 4)
seed	int	0	Rotation seed — must match at build and query time
use_prod	bool	False	Enable QJL unbiased estimator (requires bits ≥ 3)

Methods

idx.add(id, vector)                    # Add one vector
idx.add_batch(ids, vectors)            # Add N vectors (50x faster than loop)
idx.delete(id) -> bool                 # Remove by id, O(1) lookup
idx.search(query, k=10) -> list        # [(id, score), ...] descending
idx.save(path)                         # Atomic binary save
HadaMaxIndex.load(path)                # Load from .hdmx file
idx.stats() -> dict                    # Compression / memory info
len(idx)                               # Number of stored vectors

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Relation to TurboQuant / PolarQuant

hadamax implements the core compression pipeline from:

Zandieh, A., Daliri, M., Hadian, A., & Mirrokni, V. (2025). TurboQuant: Online Vector Quantization with Near-optimal Distortion Rate. ICLR 2026. arXiv:2504.19874.

The same algorithm was published concurrently under the name "PolarQuant" at AISTATS 2026. Both names were already taken as PyPI packages; hadamax = Hadamard + Lloyd-Max, named after its two core operations.

The key contributions of hadamax over the reference paper code:

• No scipy: codebooks are hardcoded, eliminating the only heavy dependency
• Batch WHT: single $O(n \cdot d \log d)$ call for bulk inserts
• Float16 cache: centroid expansions stored in half precision, ~2× faster matmul on modern hardware
• O(1) delete via _id_to_pos dict + position compaction
• Atomic persistence: .hdmx.tmp → os.replace() pattern
• Backward-compatible file format: v1/v2 both loadable

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Installation

pip install hadamax

Requirements: Python ≥ 3.10, NumPy ≥ 1.24. No other runtime dependencies.

For development:

git clone https://github.com/stffns/hadamax
cd hadamax
pip install -e ".[dev]"
pytest tests/ -v

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License

MIT © 2025 Jayson Steffens. See LICENSE.

The TurboQuant algorithm is described in arXiv:2504.19874 by Zandieh et al. (Google Research / ICLR 2026). This package is an independent implementation.