# AffineQuant: Affine Transformation Quantization for Large Language Models

arXiv: https://arxiv.org/abs/2403.12544 · HTML: https://mlsanity.com/paper/2403.12544/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 36 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [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)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [ZeroQuant: Efficient and Affordable Post-Training Quantization for Large-Scale Transformers](https://mlsanity.com/paper/2206.01861/) (2022)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
- [PACT: Parameterized Clipping Activation for Quantized Neural Networks](https://mlsanity.com/paper/1805.06085/) (2018)
- [HAQ: Hardware-Aware Automated Quantization with Mixed Precision](https://mlsanity.com/paper/1811.08886/) (2018)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning](https://mlsanity.com/paper/2208.11580/) (2022)
- [SqueezeLLM: Dense-and-Sparse Quantization](https://mlsanity.com/paper/2306.07629/) (2023)
- [Post-Training Quantization for Vision Transformer](https://mlsanity.com/paper/2106.14156/) (2021)
- [ZeroQ: A Novel Zero Shot Quantization Framework](https://mlsanity.com/paper/2001.00281/) (2020)
- [Differentiable Soft Quantization: Bridging Full-Precision and Low-Bit Neural Networks](https://mlsanity.com/paper/1908.05033/) (2019)
- [Outlier Suppression: Pushing the Limit of Low-bit Transformer Language Models](https://mlsanity.com/paper/2209.13325/) (2022)
- [QDrop: Randomly Dropping Quantization for Extremely Low-bit Post-Training Quantization](https://mlsanity.com/paper/2203.05740/) (2022)
- [Quantizable Transformers: Removing Outliers by Helping Attention Heads Do Nothing](https://mlsanity.com/paper/2306.12929/) (2023)
- [RPTQ: Reorder-based Post-training Quantization for Large Language Models](https://mlsanity.com/paper/2304.01089/) (2023)
- [Network Quantization with Element-wise Gradient Scaling](https://mlsanity.com/paper/2104.00903/) (2021)
- [Optimize Weight Rounding via Signed Gradient Descent for the Quantization of LLMs](https://mlsanity.com/paper/2309.05516/) (2023)
- [FlexRound: Learnable Rounding based on Element-wise Division for Post-Training Quantization](https://mlsanity.com/paper/2306.00317/) (2023)
- [Solving Oscillation Problem in Post-Training Quantization Through a Theoretical Perspective](https://mlsanity.com/paper/2303.11906/) (2023)

## Built on by (most cited first, 30 of 67)

- [A Survey on Efficient Inference for Large Language Models](https://mlsanity.com/paper/2404.14294/) (2024)
- [DuQuant: Distributing Outliers via Dual Transformation Makes Stronger Quantized LLMs](https://mlsanity.com/paper/2406.01721/) (2024)
- [FlatQuant: Flatness Matters for LLM Quantization](https://mlsanity.com/paper/2410.09426/) (2024)
- [A Survey on Large Language Model Acceleration based on KV Cache Management](https://mlsanity.com/paper/2412.19442/) (2024)
- [OstQuant: Refining Large Language Model Quantization with Orthogonal and Scaling Transformations for Better Distribution Fitting](https://mlsanity.com/paper/2501.13987/) (2025)
- [SliM-LLM: Salience-Driven Mixed-Precision Quantization for Large Language Models](https://mlsanity.com/paper/2405.14917/) (2024)
- [Quantization Error Propagation: Revisiting Layer-Wise Post-Training Quantization](https://mlsanity.com/paper/2504.09629/) (2025)
- [SpikeLLM: Scaling up Spiking Neural Network to Large Language Models via Saliency-based Spiking](https://mlsanity.com/paper/2407.04752/) (2024)
- [MagR: Weight Magnitude Reduction for Enhancing Post-Training Quantization](https://mlsanity.com/paper/2406.00800/) (2024)
- [Block Rotation is All You Need for MXFP4 Quantization](https://mlsanity.com/paper/2511.04214/) (2025)
- [AMXFP4: Taming Activation Outliers with Asymmetric Microscaling Floating-Point for 4-bit LLM Inference](https://mlsanity.com/paper/2411.09909/) (2024)
- [Quantized Visual Geometry Grounded Transformer](https://mlsanity.com/paper/2509.21302/) (2025)
- [DartQuant: Efficient Rotational Distribution Calibration for LLM Quantization](https://mlsanity.com/paper/2511.04063/) (2025)
- [MixLLM: LLM Quantization with Global Mixed-precision between Output-features and Highly-efficient System Design](https://mlsanity.com/paper/2412.14590/) (2024)
- [QSVD: Efficient Low-rank Approximation for Unified Query-Key-Value Weight Compression in Low-Precision Vision-Language Models](https://mlsanity.com/paper/2510.16292/) (2025)
- [Matryoshka Quantization](https://mlsanity.com/paper/2502.06786/) (2025)
- [SocialOmni: Benchmarking Audio-Visual Social Interactivity in Omni Models](https://mlsanity.com/paper/2603.16859/) (2026)
- [BoA: Attention-aware Post-training Quantization without Backpropagation](https://mlsanity.com/paper/2406.13474/) (2024)
- [SLiM: One-shot Quantization and Sparsity with Low-rank Approximation for LLM Weight Compression](https://mlsanity.com/paper/2410.09615/) (2024)
- [Benchmarking Post-Training Quantization in LLMs: Comprehensive Taxonomy, Unified Evaluation, and Comparative Analysis](https://mlsanity.com/paper/2502.13178/) (2025)
- [Identifying Sensitive Weights via Post-quantization Integral](https://mlsanity.com/paper/2503.01901/) (2025)
- [Low-bit Model Quantization for Deep Neural Networks: A Survey](https://mlsanity.com/paper/2505.05530/) (2025)
- [ButterflyQuant: Ultra-low-bit LLM Quantization through Learnable Orthogonal Butterfly Transforms](https://mlsanity.com/paper/2509.09679/) (2025)
- [What Makes Low-Bit Quantization-Aware Training Work for Reasoning LLMs? A Systematic Study](https://mlsanity.com/paper/2601.14888/) (2026)
- [DuQuant++: Fine-grained Rotation Enhances Microscaling FP4 Quantization](https://mlsanity.com/paper/2604.17789/) (2026)
- [HadaNorm: Diffusion Transformer Quantization through Mean-Centered Transformations](https://mlsanity.com/paper/2506.09932/) (2025)
- [QuantSparse: Comprehensively Compressing Video Diffusion Transformer with Model Quantization and Attention Sparsification](https://mlsanity.com/paper/2509.23681/) (2025)
- [MorphServe: Efficient and Workload-Aware LLM Serving via Runtime Quantized Layer Swapping and KV Cache Resizing](https://mlsanity.com/paper/2506.02006/) (2025)
- [SINQ: Sinkhorn-Normalized Quantization for Calibration-Free Low-Precision LLM Weights](https://mlsanity.com/paper/2509.22944/) (2025)
- [BATQuant: Outlier-resilient MXFP4 Quantization via Learnable Block-wise Optimization](https://mlsanity.com/paper/2603.16590/) (2026)

## Abstract (arXiv)

The significant resource requirements associated with Large-scale Language Models (LLMs) have generated considerable interest in the development of techniques aimed at compressing and accelerating neural networks. Among these techniques, Post-Training Quantization (PTQ) has emerged as a subject of considerable interest due to its noteworthy compression efficiency and cost-effectiveness in the context of training. Existing PTQ methods for LLMs limit the optimization scope to scaling transformations between pre- and post-quantization weights. In this paper, we advocate for the direct optimization using equivalent Affine transformations in PTQ (AffineQuant). This approach extends the optimization scope and thus significantly minimizing quantization errors. Additionally, by employing the corresponding inverse matrix, we can ensure equivalence between the pre- and post-quantization outputs of PTQ, thereby maintaining its efficiency and generalization capabilities. To ensure the invertibility of the transformation during optimization, we further introduce a gradual mask optimization method. This method initially focuses on optimizing the diagonal elements and gradually extends to the other elements. Such an approach aligns with the Levy-Desplanques theorem, theoretically ensuring invertibility of the transformation. As a result, significant performance improvements are evident across different LLMs on diverse datasets. To illustrate, we attain a C4 perplexity of 15.76 (2.26 lower vs 18.02 in OmniQuant) on the LLaMA2-7B model of W4A4 quantization without overhead. On zero-shot tasks, AffineQuant achieves an average of 58.61 accuracy (1.98 lower vs 56.63 in OmniQuant) when using 4/4-bit quantization for LLaMA-30B, which setting a new state-of-the-art benchmark for PTQ in LLMs.