# Transforms for LLM Quantization: The Great Inversion and Format Co-Design

arXiv: https://arxiv.org/abs/2608.25188 · HTML: https://mlsanity.com/paper/2608.25188/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 56 of its 200 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 30 of 56)

- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [ASVD: Activation-aware Singular Value Decomposition for Compressing Large Language Models](https://mlsanity.com/paper/2312.05821/) (2023)
- [Microscaling Data Formats for Deep Learning](https://mlsanity.com/paper/2310.10537/) (2023)
- [TurboQuant: Online Vector Quantization with Near-optimal Distortion Rate](https://mlsanity.com/paper/2504.19874/) (2025)
- [RPTQ: Reorder-based Post-training Quantization for Large Language Models](https://mlsanity.com/paper/2304.01089/) (2023)
- [GPTVQ: The Blessing of Dimensionality for LLM Quantization](https://mlsanity.com/paper/2402.15319/) (2024)
- [Pretraining Large Language Models with NVFP4](https://mlsanity.com/paper/2509.25149/) (2025)
- [Four Over Six: More Accurate NVFP4 Quantization with Adaptive Block Scaling](https://mlsanity.com/paper/2512.02010/) (2025)
- [A Survey of Low-bit Large Language Models: Basics, Systems, and Algorithms](https://mlsanity.com/paper/2409.16694/) (2024)
- [Qronos: Correcting the Past by Shaping the Future... in Post-Training Quantization](https://mlsanity.com/paper/2505.11695/) (2025)
- [PolarQuant: Quantizing KV Caches with Polar Transformation](https://mlsanity.com/paper/2502.02617/) (2025)
- [NestQuant: Nested Lattice Quantization for Matrix Products and LLMs](https://mlsanity.com/paper/2502.09720/) (2025)
- [Pushing the Limits of Large Language Model Quantization via the Linearity Theorem](https://mlsanity.com/paper/2411.17525/) (2024)
- [Quartet II: Accurate LLM Pre-Training in NVFP4 by Improved Unbiased Gradient Estimation](https://mlsanity.com/paper/2601.22813/) (2026)
- [Block Rotation is All You Need for MXFP4 Quantization](https://mlsanity.com/paper/2511.04214/) (2025)
- [INT v.s. FP: A Comprehensive Study of Fine-Grained Low-bit Quantization Formats](https://mlsanity.com/paper/2510.25602/) (2025)
- [Provable Post-Training Quantization: Theoretical Analysis of OPTQ and Qronos](https://mlsanity.com/paper/2508.04853/) (2025)
- [QQQ: Quality Quattuor-Bit Quantization for Large Language Models](https://mlsanity.com/paper/2406.09904/) (2024)
- [Optimal Quantization for Matrix Multiplication](https://mlsanity.com/paper/2410.13780/) (2024)
- [NeuZip: Memory-Efficient Training and Inference with Dynamic Compression of Neural Networks](https://mlsanity.com/paper/2410.20650/) (2024)
- [HadaCore: Tensor Core Accelerated Hadamard Transform Kernel](https://mlsanity.com/paper/2412.08832/) (2024)
- [LRQ-DiT: Log-Rotation Post-Training Quantization of Diffusion Transformers for Image and Video Generation](https://mlsanity.com/paper/2508.03485/) (2025)
- [Pushing the Limits of Block Rotations in Post-Training Quantization](https://mlsanity.com/paper/2601.22347/) (2026)
- [HiFloat4 Format for Language Model Inference](https://mlsanity.com/paper/2602.11287/) (2026)
- [ButterflyQuant: Ultra-low-bit LLM Quantization through Learnable Orthogonal Butterfly Transforms](https://mlsanity.com/paper/2509.09679/) (2025)
- [KVLinC : KV Cache Quantization with Hadamard Rotation and Linear Correction](https://mlsanity.com/paper/2510.05373/) (2025)
- [Benchmarking Post-Training Quantization of Large Language Models under Microscaling Floating Point Formats](https://mlsanity.com/paper/2601.09555/) (2026)
- [OSCAR: Offline Spectral Covariance-Aware Rotation for 2-bit KV Cache Quantization](https://mlsanity.com/paper/2605.17757/) (2026)
- [TEQ: Trainable Equivalent Transformation for Quantization of LLMs](https://mlsanity.com/paper/2310.10944/) (2023)
- [Huff-LLM: End-to-End Lossless Compression for Efficient LLM Inference](https://mlsanity.com/paper/2502.00922/) (2025)

## Abstract (arXiv)

Most competitive 4-bit LLM research pipelines now open the same way: apply a linear, function-preserving transform (rotation, scaling, permutation, non-orthogonal affine) so the outlier mass sits more favorably against the group scales, and only then round. Yet we are aware of no survey dedicated to this transform stage, and its literature is quietly re-deriving an older theory. We identify and formalize the principle that organizes it, the Great Inversion: allocation-flexible coding rewards energy concentration, whereas the grouped shared-scale quantization a deployed matrix instruction performs rewards within-group flattening. Classical transform coding (1963: decorrelate, allocate bits, quantize) spends different bits per coordinate at a fixed total rate; for a Gaussian source at high rate the Karhunen-Loeve transform's concentration minimizes distortion. A deployed operand tile instead carries one absolute-maximum scale per group and equal bits everywhere, with no allocation; on a uniform grid that objective rewards flattening, approached by Hadamard incoherence. We prove that opposition under within-group majorization: the prescriptions point in opposite directions, each backed by a proof against its own objective, and for a generic spectrum no optimality guarantee transfers. A second axis is the number format: the non-uniform FP4 grid makes flattening buy less, MXFP4's power-of-two block scale still rewards a rotation confined to that block, and NVFP4's mantissa-carrying scale largely removes that pull, so the target pole depends jointly on allocation regime and format. We survey 200 works to a June 2026 cutoff; classify 43 transform methods by structure, data-awareness, searched-versus-constructed, and runtime cost; record, where reported, how they compose with GPTQ rounding; distill a first-choice guide by deployment regime; and close with the open problems it exposes.