# PCDVQ: Enhancing Vector Quantization for Large Language Models via Polar Coordinate Decoupling

arXiv: https://arxiv.org/abs/2506.05432 · HTML: https://mlsanity.com/paper/2506.05432/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 37 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 27 of 27)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [ZeroQuant: Efficient and Affordable Post-Training Quantization for Large-Scale Transformers](https://mlsanity.com/paper/2206.01861/) (2022)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [Extreme Compression of Large Language Models via Additive Quantization](https://mlsanity.com/paper/2401.06118/) (2024)
- [OstQuant: Refining Large Language Model Quantization with Orthogonal and Scaling Transformations for Better Distribution Fitting](https://mlsanity.com/paper/2501.13987/) (2025)
- [QTIP: Quantization with Trellises and Incoherence Processing](https://mlsanity.com/paper/2406.11235/) (2024)
- [GPTVQ: The Blessing of Dimensionality for LLM Quantization](https://mlsanity.com/paper/2402.15319/) (2024)
- [WKVQuant: Quantizing Weight and Key/Value Cache for Large Language Models Gains More](https://mlsanity.com/paper/2402.12065/) (2024)
- [VPTQ: Extreme Low-bit Vector Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2409.17066/) (2024)
- [DB-LLM: Accurate Dual-Binarization for Efficient LLMs](https://mlsanity.com/paper/2402.11960/) (2024)
- [PolarQuant: Quantizing KV Caches with Polar Transformation](https://mlsanity.com/paper/2502.02617/) (2025)
- [PolarQuant: Leveraging Polar Transformation for Efficient Key Cache Quantization and Decoding Acceleration](https://mlsanity.com/paper/2502.00527/) (2025)

## Built on by (most cited first, 7 of 7)

- [KBVQ-MoE: KLT-guided SVD with Bias-Corrected Vector Quantization for MoE Large Language Models](https://mlsanity.com/paper/2602.11184/) (2026)
- [RSAVQ: Riemannian Sensitivity-Aware Vector Quantization for Large Language Models](https://mlsanity.com/paper/2510.01240/) (2025)
- [MGVQ: Synergizing Multi-dimensional Sensitivity-Aware and Gradient-Hessian Fusion for Vector Quantization](https://mlsanity.com/paper/2605.24019/) (2026)
- [SPHQuant: Efficient extreme low bit weight quantization for Vision-Language Models](https://mlsanity.com/paper/2609.24875/) (2026)
- [Embedding Compression via Spherical Coordinates](https://mlsanity.com/paper/2602.00079/) (2026)
- [Transforms for LLM Quantization: The Great Inversion and Format Co-Design](https://mlsanity.com/paper/2608.25188/) (2026)
- [SAGE: Pushing 2-Bit Vector-Quantized LLMs toward Real-World Usability](https://mlsanity.com/paper/2610.14745/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) face significant challenges in edge deployment due to their massive parameter scale. Vector Quantization (VQ), a clustering-based quantization method, serves as a prevalent solution to this issue for its extremely low-bit (even at 2-bit) and considerable accuracy. Since a vector is a quantity in mathematics and physics that has both direction and magnitude, existing VQ works typically quantize them in a coupled manner. However, we find that direction exhibits significantly greater sensitivity to quantization compared to the magnitude. For instance, when separately clustering the directions and magnitudes of weight vectors in LLaMA-2-7B, the accuracy drop of zero-shot tasks are 46.5% and 2.3%, respectively. This gap even increases with the reduction of clustering centers. Further, Euclidean distance, a common metric to access vector similarities in current VQ works, places greater emphasis on reducing the magnitude error. This property is contrary to the above finding, unavoidably leading to larger quantization errors. To these ends, this paper proposes Polar Coordinate Decoupled Vector Quantization (PCDVQ), an effective and efficient VQ framework consisting of two key modules: 1) Polar Coordinate Decoupling (PCD), which transforms vectors into their polar coordinate representations and perform independent quantization of the direction and magnitude parameters.2) Distribution Aligned Codebook Construction (DACC), which optimizes the direction and magnitude codebooks in accordance with the source distribution. Experimental results show that PCDVQ outperforms baseline methods at 2-bit level by at least 1.5% zero-shot accuracy, establishing a novel paradigm for accurate and highly compressed LLMs.