glq 0.2.2

E8 lattice codebook quantization for LLM weights

GLQ

Post-training weight quantization for LLMs using E8 lattice codebooks.

GLQ encodes weights into 8-dimensional E8 lattice points via nearest-neighbor lookup. A Randomized Hadamard Transform (RHT) makes the Hessian approximately diagonal so that Euclidean nearest-neighbor is near-optimal under the proxy loss.

Results

SmolLM3-3B-Base on WikiText-2 (128 calibration samples, NVIDIA L40S):

Method	Eff. BPW	Size (MB)	Perplexity	vs bf16
bf16	16.00	6150	7.04	1.00x
GLQ 4-bit	4.00	2531	7.19	1.02x
AWQ 4-bit†	5.60	2152	8.15	1.16x
QuIP+GPTQ 4-bit†	4.76	1829	8.17	1.16x
GLQ 3.5-bit mixed	3.50	2282	7.20	1.02x
GLQ 3-bit	3.00	2531	7.64	1.09x
GLQ 3-bit mixed (2+4)	3.00	2035	7.65	1.09x
GLQ 2.5-bit mixed	2.50	2031	8.08	1.15x
QuIP+GPTQ 3-bit†	3.70	1423	9.30	1.32x
GLQ 2-bit	2.00	1531	9.61	1.36x

†AWQ and QuIP+GPTQ numbers are from an earlier measurement on A10G with 16 calibration samples and may not be directly comparable. GLQ numbers were re-measured on L40S with 128 samples.

Mixed-precision models use --bpw <target> --min-bpw 2 --max-bpw 4 to automatically allocate per-layer bit widths via Hessian sensitivity profiling. Sensitive layers (attention projections in the middle of the network) get higher precision while robust layers are compressed further. GLQ 3.5-bit mixed matches uniform 4-bit quality (7.20 vs 7.19) at 10% less storage.

Mistral-7B-v0.3 on WikiText-2 (16 calibration samples, NVIDIA A10G):

Method	BPW	Perplexity	vs bf16	GPU MB	tok/s
bf16	16	4.20	1.00x	14505	28.1
GLQ 3-bit	3	4.41	1.05x	4436	9.7

Ministral-3-3B-Base-2512 on WikiText-2 (16 calibration samples, NVIDIA A10G):

Method	BPW	Perplexity	vs bf16	GPU MB	tok/s
bf16	16	5.91	1.00x	7348	37.0
GLQ 3-bit	3	6.47	1.09x	3788	11.4

Nemotron-3-Nano-30B (hybrid Mamba-Attention-MoE) on WikiText-2 (16 calibration samples, NVIDIA A10G):

Method	BPW	Perplexity	vs bf16
bf16	16	7.72	1.00x
GLQ 3-bit	3	8.09	1.05x

Llama-3.2-3B on WikiText-2 (16 calibration samples, NVIDIA A10G):

Method	BPW	Perplexity	vs bf16	GPU MB	tok/s
bf16	16	6.17	1.00x	6137	37.6
GLQ 3-bit	3	6.78	1.10x	3529	10.8
GLQ 2-bit	2	8.49	1.38x	3526	11.0

SmolLM3-3B-Base 5-task accuracy via lm-evaluation-harness (acc_norm where available, 128 calibration samples, NVIDIA L40S):

Method	Eff. BPW	ARC-c	ARC-e	HellaSwag	PIQA	WinoGrande	Avg
bf16 baseline	16.00	0.540	0.793	0.758	0.786	0.668	0.709
GLQ 3-bit	3.00	0.447	0.759	0.530	0.755	0.688	0.636
GLQ 2-bit	2.00	0.415	0.679	0.634	0.730	0.660	0.623

GLQ 2-bit retains 87.9% of bf16 accuracy at 8x compression (exactly 2.00 effective bpw, no group scales).

SmolLM3-3B-Base 4-bit method comparison (acc_norm where available, 128 calibration samples, NVIDIA L40S):

Method	Eff. BPW	ARC-c	ARC-e	HellaSwag	PIQA	WinoGrande	Avg	tok/s	VRAM
bf16 baseline	16.00	0.540	0.793	0.758	0.786	0.668	0.709	30.8	5,875 MB
AutoRound W4	4.50	0.532	0.797	0.748	0.781	0.661	0.704	—	—
GPTQ W4A16	4.50	0.538	0.785	0.740	0.781	0.648	0.698	8.5†	7,487 MB
GLQ 4-bit	4.00	0.522	0.776	0.746	0.780	0.672	0.699	14.0	5,044 MB

GLQ 4-bit retains 98.6% of bf16 accuracy at exactly 4.00 effective bpw (no group scales). AutoRound (99.3%) and GPTQ (98.5%) use group_size=128 (~4.5 eff bpw). Decode tok/s at B=1 steady-state on NVIDIA L40S. †GPTQ without compiled CUDA dequant kernels (auto-gptq/gptqmodel); native kernels would be faster.

SmolLM2-360M on WikiText-2 (128 calibration samples, NVIDIA L40S):

Method	Eff. BPW	Perplexity	vs bf16
bf16 baseline	16.00	11.47	1.00x
GLQ 4-bit	4.00	11.77	1.03x
QuIP+GPTQ 4-bit	4.75	12.06	1.05x
GLQ 3-bit	3.00	13.16	1.15x
QuIP+GPTQ 3-bit	3.69	14.84	1.29x
GLQ 2-bit	2.00	17.94	1.56x
GPTQ 3-bit	9.48	18.61	1.62x

GLQ uses a single global scale per layer rather than per-group scales, so effective bit widths match the nominal rate exactly. GLQ 2-bit (17.94) beats GPTQ 3-bit (18.61) at less than 1/4 the storage. GLQ 4-bit (11.77) beats QuIP+GPTQ 4-bit (12.06) at lower effective bpw (4.00 vs 4.75).

Inference performance

SmolLM3-3B-Base prefill throughput on NVIDIA L40S (seqlen=128):

Batch	GLQ tok/s	bf16 tok/s	Speedup	GLQ VRAM	bf16 VRAM
1	1,160	4,709	0.25x	3,832 MB	6,194 MB
4	3,060	18,845	0.16x	4,185 MB	6,294 MB
64	2,394	26,528	0.09x	6,611 MB	8,295 MB

GLQ compresses model weights from 6.2 GB to 2.5 GB (60% reduction). Runtime fp32 intermediates from the RHT add ~1.3 GB overhead, giving a net VRAM savings of ~38% at B=1. The throughput gap is due to codebook scatter-gather in the dequant kernel vs contiguous cuBLAS reads; closing this requires CUDA C kernels with warp shuffles and async prefetch.

How it works

1. E8 lattice codebook: 65536 vectors from the first 7 shells of the E8 lattice. Each 8-weight group maps to a 16-bit index (2 bpw). For 3/4 bpw, a second-stage residual codebook adds 8 or 16 more bits.

2. Randomized Hadamard Transform (RHT): Random sign flips + Fast Walsh-Hadamard Transform applied to both weights and Hessian. This spreads weight magnitude evenly across dimensions, making the Hessian block-diagonal approximately proportional to identity. After RHT, Euclidean nearest-neighbor in the codebook is close to Hessian-optimal.

3. LDLQ error feedback: Block-LDL decomposition of the Hessian drives a sequential quantization sweep (like GPTQ but over 8-dim blocks instead of scalar columns). Quantization error from each block propagates forward to correct subsequent blocks.

4. Fused Triton inference kernel: On CUDA, a custom Triton kernel reads codebook indices directly from HBM and gathers from the L2-cached codebook (65536 x 8 fp16 = 1 MB) without ever materializing the full weight matrix. This provides real GPU memory savings proportional to the compression ratio.

Install

Requires Python 3.10+ and PyTorch 2.0+. Install PyTorch first (pytorch.org), then:

# Full install (includes transformers, datasets, etc. for glq-quantize CLI):
pip install 'glq[quantize]'

# Or minimal install (inference only, no quantization dependencies):
pip install glq

Triton is bundled with PyTorch on CUDA and will be used automatically when available. On CPU, GLQ falls back to a naive dequantize-then-matmul path.

Quickstart

Quantizing a model

Command line

# 2-bit quantization (smallest model, ~1.5x perplexity)
glq-quantize \
    --model HuggingFaceTB/SmolLM2-360M \
    --output ./smollm2-glq-2bpw \
    --bpw 2 \
    --nsamples 128 \
    --device cuda

# 3-bit quantization (good balance of size and quality)
glq-quantize \
    --model HuggingFaceTB/SmolLM2-360M \
    --output ./smollm2-glq-3bpw \
    --bpw 3 \
    --nsamples 128 \
    --device cuda

# 4-bit quantization (near-lossless, ~1.03x perplexity)
glq-quantize \
    --model HuggingFaceTB/SmolLM2-360M \
    --output ./smollm2-glq-4bpw \
    --bpw 4 \
    --nsamples 128 \
    --device cuda

All CLI options:

glq-quantize --help
  --model              HuggingFace model ID or local path (required)
  --output             Output directory for quantized model (required)
  --bpw                Bits per weight: 2, 3, or 4 (default: 2)
  --tune-iters         LDLQ refinement iterations (default: 0)
  --nsamples           Calibration samples from WikiText-2 (default: 16)
  --seqlen             Calibration sequence length (default: 2048)
  --device             cuda or cpu (default: cuda)
  --trust-remote-code  Allow custom model code from HF Hub
  --streaming          Load weights layer-by-layer from safetensors
                       (for models exceeding system RAM)

Python API

from glq import quantize

quantize(
    model_name="HuggingFaceTB/SmolLM2-360M",
    output_dir="./smollm2-glq-4bpw",
    bpw=4,
    nsamples=128,
    device="cuda",
)

The quantize() function handles the full pipeline: load model, capture Hessians via calibration data, quantize each linear layer with E8+RHT+LDLQ, and save the result as a standard HuggingFace model directory (safetensors + config.json + tokenizer).

Loading and running a quantized model

import glq.hf_integration  # registers GLQ with transformers
from transformers import AutoModelForCausalLM, AutoTokenizer

model = AutoModelForCausalLM.from_pretrained(
    "./smollm2-glq-4bpw",
    device_map="auto",
)
tokenizer = AutoTokenizer.from_pretrained("./smollm2-glq-4bpw")

inputs = tokenizer("The capital of France is", return_tensors="pt").to(model.device)
output = model.generate(**inputs, max_new_tokens=50)
print(tokenizer.decode(output[0], skip_special_tokens=True))

The import glq.hf_integration line registers GLQ as a quantization method with HuggingFace Transformers. After that, from_pretrained automatically:

1. Reads quantization_config.quant_method = "glq" from config.json
2. Replaces nn.Linear modules with E8RHTLinear
3. Loads the quantized weights (codebook indices + sign vectors)
4. Builds the E8 codebook and attaches it to all quantized layers

On CUDA, inference automatically uses the fused Triton kernel. On CPU, it falls back to dequantize-then-matmul.

Bit widths

BPW	Encoding	Bits per 8 weights	Storage
2	16-bit codebook index	16	Global scale only
3	16-bit primary + 8-bit residual index	24	Global scale + residual scale
4	16-bit primary + 16-bit residual index	32	Global scale + residual scale

All bit widths use a single global scale per layer (no group-size parameter), so effective bit widths match the nominal rate exactly.

For 3/4 bpw, GLQ uses a two-stage residual vector quantization (RVQ): the primary codebook (65536 entries) encodes the bulk of the weight, and a secondary codebook (256 entries for 3 bpw, 65536 for 4 bpw) encodes the residual error scaled by a learned factor.

Triton inference kernel

The fused Triton kernel (glq/inference_kernel.py) is the core of GLQ's inference performance. It computes Y = X @ dequant(W)^T without materializing the full weight matrix.

How it works

Instead of the naive approach (decode all indices into a dense bf16 matrix, then matmul), the kernel:

1. Iterates over N/8 codebook blocks per output row
2. Loads int16 indices from HBM and gathers 8-element vectors from the L2-cached codebook
3. Accumulates dot products (matvec) or Tensor Core matmuls (prefill) against the gathered codebook vectors
4. Applies the global scale factor and writes the output

This means GPU memory holds only the compressed indices (2 bytes per 8 weights) rather than the full fp16 weight matrix (16 bytes per 8 weights) — an 8x reduction at 2 bpw.

Kernel variants

• Tensor Core matmul kernel (_glq_dequant_matmul_tc_kernel): For batch sizes >= 2 (prefill). Processes pairs of codebook blocks to form K=16 tiles for tl.dot (mma.m16n8k16 Tensor Core instructions). Autotuned over BLOCK_B and BLOCK_M.
• Matvec kernel (_glq_dequant_matvec_kernel): For B=1 (autoregressive decode). Autotuned over BLOCK_M and num_warps for the memory-bound single-token case.
• Fused RHT kernels (_input_rht_kernel, _output_rht_kernel): Fuse pad + sign vector + Fast Hadamard Transform into single kernel launches, eliminating per-layer Python overhead.

All kernels support two-stage RVQ for 3/4 bpw via a HAS_STAGE2 compile-time constant.

Using the kernel directly

The kernel is used automatically by E8RHTLinear.forward() when running on CUDA with Triton available. You can also call it directly:

from glq.inference_kernel import glq_dequant_matmul

# 2bpw: single codebook
y = glq_dequant_matmul(
    x,          # (B, N) input activations, fp16/fp32
    Qidxs,      # (M, N//8) codebook indices, int16
    codebook,    # (65536, 8) codebook vectors, fp16
    Wscale,      # float, global scale factor
)

# 3/4bpw: two-stage with residual codebook
y = glq_dequant_matmul(
    x, Qidxs, codebook, Wscale,
    Qidxs2=Qidxs2,              # (M, N//8) secondary indices, int16
    codebook2=codebook2,          # (K2, 8) secondary codebook, fp16
    inv_resid_scale=inv_rs,       # float, 1.0 / residual_scale
)

Falls back to naive dequantize+matmul on CPU or when Triton is not available.

Requirements

• CUDA GPU
• Triton (bundled with pip install torch on CUDA, or pip install 'glq[cuda]')
• PyTorch 2.0+

Architecture

glq/
  codebook.py          # E8ShellCodebook: enumeration, encode/decode, make_small()
  hadamard.py          # Fast Walsh-Hadamard Transform
  rht.py               # Randomized Hadamard Transform (sign flips + FHT)
  ldlq.py              # Block-LDL quantization with error feedback
  quantize_model.py    # Full model quantization pipeline + CLI
  quantized_linear.py  # E8RHTLinear: drop-in nn.Linear replacement
  inference_kernel.py  # Fused Triton dequant+matmul kernels
  hf_integration.py    # HuggingFace Transformers integration

Acknowledgments

• The RHT incoherence approach follows QuIP# (Tseng et al., 2024)
• E8 lattice geometry from Conway & Sloane, Sphere Packings, Lattices and Groups
• LDLQ error feedback from GPTQ (Frantar et al., 2022)

License

Apache 2.0