Initialisation Determines the Basin: Efficient Codebook Optimisation for Extreme LLM Quantization

As large language models (LLMs) grow in parameter count, model storage and parameter memory traffic have become major bottlenecks to efficient deployment. Codebook-based weight quantization reduces these costs, but imbalanced nearest-codeword assignments during fitting can leave some codewords insufficiently updated, limiting effective codebook utilization. To address this limitation, we propose Balanced Assignment Refinement for Quantization (BARQ), which improves quantization quality through balanced fitting of existing codebooks. Specifically, we first compute joint soft assignments between weight blocks and codewords through entropically regularized optimal transport with uniform marginals and curvature-weighted reconstruction costs, ensuring equal positive fitting mass for every codeword in the exact solution. We then refine the codewords through an assignment-weighted barycentric update, which we prove minimizes the fitting objective for fixed assignments. For finite Sinkhorn iterations, the implemented update retains this optimality provided all codeword masses exceed the denominator floor. Finally, we discard the soft assignments and use the refined codebook for standard hard nearest-codeword encoding, with our analysis establishing sufficient conditions for reducing hard-quantization distortion and evaluation loss. Across multiple LLMs, BARQ achieves lower perplexity and higher mean zero-shot accuracy than the evaluated baselines at comparable bit budgets. The code is available at https://github.com/chenhangcuisg-code/BARQ.

Extreme low-bit LLM compression is most challenging when weights, activations, and KV caches are quantized together: their distributions differ, and quantization errors interact throughout the network. We introduce CanonQ, a unified quantization-aware training framework that addresses these challenges by separating source canonicalization from task-aware adaptation. Fixed rotations and energy normalization map heterogeneous tensor sources to canonical coordinates, enabling frozen Gaussian-reference codebooks to be reused across layers and models. Joint training then adapts the network to the coupled errors of weight, activation, and cache quantization within a common scalar/vector interface. We bound frozen-codebook transfer error and local task loss, and derive an exact normalization-aware straight-through Jacobian that links quantization distortion to gradient bias. The strongest gains arise under joint W2A4KV2 compression: across LLaMA3-1B/3B/8B, CanonQ-Omni achieves up to 14.28x lower WikiText-2 perplexity and up to 57.9% higher mean zero-shot accuracy than prior state-of-the-art and representative quantization baselines. The benefits extend to Qwen3-1.7B, code generation, and mathematical reasoning: on instruction-tuned MobileLLM-Pro-1B at W2A16KV16, CanonQ achieves relative improvements of 41.7% in HumanEval pass@1 and 39.1% in GSM8K exact match over the strongest evaluated quantization baseline.

We present PolarQuant, a post-training weight quantization method for large language models (LLMs) that exploits the distributional structure of neural network weights to achieve near-lossless compression. PolarQuant operates in three stages: (1) block-wise normalization to the unit hypersphere, (2) Walsh-Hadamard rotation to transform coordinates into approximately Gaussian random variables, and (3) quantization with centroids matched to the Gaussian distribution. Our ablation reveals that Hadamard rotation alone accounts for 98% of the quality improvement, reducing Qwen3.5-9B perplexity from 6.90 (absmax Q5) to 6.40 (Delta = +0.03 from FP16), making it practically lossless without any calibration data. Furthermore, PolarQuant functions as an effective preprocessing step for downstream INT4 quantizers: PolarQuant Q5 dequantized and re-quantized by torchao INT4 achieves perplexity 6.56 versus 6.68 for direct absmax INT4, while maintaining 43.1 tok/s throughput at 6.5 GB VRAM. Code and models are publicly available.