# TORQUE: Optimizing What (not) to Quantize Before and After Rotation

arXiv: https://arxiv.org/abs/2609.36032 · HTML: https://mlsanity.com/paper/2609.36032/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 44 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 30 of 34)

- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [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)
- [SpQR: A Sparse-Quantized Representation for Near-Lossless LLM Weight Compression](https://mlsanity.com/paper/2306.03078/) (2023)
- [Distributed Mean Estimation with Limited Communication](https://mlsanity.com/paper/1611.00429/) (2016)
- [TurboQuant: Online Vector Quantization with Near-optimal Distortion Rate](https://mlsanity.com/paper/2504.19874/) (2025)
- [OWQ: Outlier-Aware Weight Quantization for Efficient Fine-Tuning and Inference of Large Language Models](https://mlsanity.com/paper/2306.02272/) (2023)
- [EDEN: Communication-Efficient and Robust Distributed Mean Estimation for Federated Learning](https://mlsanity.com/paper/2108.08842/) (2021)
- [DRIVE: One-bit Distributed Mean Estimation](https://mlsanity.com/paper/2105.08339/) (2021)
- [RaBitQ: Quantizing High-Dimensional Vectors with a Theoretical Error Bound for Approximate Nearest Neighbor Search](https://mlsanity.com/paper/2405.12497/) (2024)
- [DoCoFL: Downlink Compression for Cross-Device Federated Learning](https://mlsanity.com/paper/2302.00543/) (2023)
- [Practical and Asymptotically Optimal Quantization of High-Dimensional Vectors in Euclidean Space for Approximate Nearest Neighbor Search](https://mlsanity.com/paper/2409.09913/) (2024)
- [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)
- [THC: Accelerating Distributed Deep Learning Using Tensor Homomorphic Compression](https://mlsanity.com/paper/2302.08545/) (2023)
- [How to send a real number using a single bit (and some shared randomness)](https://mlsanity.com/paper/2010.02331/) (2020)
- [HACK: Homomorphic Acceleration via Compression of the Key-Value Cache for Disaggregated LLM Inference](https://mlsanity.com/paper/2502.03589/) (2025)
- [OptiReduce: Resilient and Tail-Optimal AllReduce for Distributed Deep Learning in the Cloud](https://mlsanity.com/paper/2310.06993/) (2023)
- [A Note on TurboQuant and the Earlier DRIVE/EDEN Line of Work](https://mlsanity.com/paper/2604.18555/) (2026)
- [Provable Quantization with Randomized Hadamard Transform](https://mlsanity.com/paper/2605.13810/) (2026)
- [Beyond Throughput and Compression Ratios: Towards High End-to-end Utility of Gradient Compression](https://mlsanity.com/paper/2407.01378/) (2024)
- [Quantizing With Randomized Hadamard Transforms: Efficient Heuristic Now Proven](https://mlsanity.com/paper/2605.06014/) (2026)
- [RaanA: A Fast, Flexible, and Data-Efficient Post-Training Quantization Algorithm](https://mlsanity.com/paper/2504.03717/) (2025)
- [DynamiQ: Accelerating Gradient Synchronization using Compressed Multi-hop All-reduce](https://mlsanity.com/paper/2602.08923/) (2026)
- [OptiNIC: A Resilient and Tail-Optimal RDMA NIC for Distributed ML Workloads](https://mlsanity.com/paper/2512.22743/) (2025)

## Abstract (arXiv)

Uniform random rotations are an effective preprocessing step for quantization: they make normalized coordinate distributions approximately Gaussian, enabling the use of codebooks optimized offline. We introduce TORQUE, a framework that improves on previous quantization works that use random rotations by jointly optimizing how many and which coordinates to preserve at high precision both before and after rotation, under a fixed overall expected bit budget. Intuitively, before rotation, preserving large input coordinates at high precision can reduce overall error by preventing the rotation from spreading their values across many coordinates. Likewise, after rotation, preserving a small fraction of the largest-magnitude coordinates at high precision allows the remaining values to be quantized more accurately using codebooks optimized offline for the resulting truncated Gaussian distribution. We derive a quantization error upper bound and prove that top-k pre-rotation retention minimizes it for each k. This reduces the search over coordinate subsets to an optimization over k, enabling a fast optimizer that uses offline codebooks and parallel parameter selection for practical implementation. We demonstrate an improved tradeoff between reconstruction accuracy and storage cost through numerical evaluation under the Gaussian model and experiments on nearest-neighbor retrieval, KV-cache compression, and activation compression.