# Quantization Error Is Spectrally Flat: A Single Random Probe Is a Calibrated, Data-Free Sensitivity Estimator, with Application to Budget-Targeted Mixed-Precision Quantization

arXiv: https://arxiv.org/abs/2609.33923 · HTML: https://mlsanity.com/paper/2609.33923/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 40 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [HAWQ: Hessian AWare Quantization of Neural Networks with Mixed-Precision](https://mlsanity.com/paper/1905.03696/) (2019)
- [SpQR: A Sparse-Quantized Representation for Near-Lossless LLM Weight Compression](https://mlsanity.com/paper/2306.03078/) (2023)
- [SqueezeLLM: Dense-and-Sparse Quantization](https://mlsanity.com/paper/2306.07629/) (2023)
- [HAWQ-V2: Hessian Aware trace-Weighted Quantization of Neural Networks](https://mlsanity.com/paper/1911.03852/) (2019)
- [Extreme Compression of Large Language Models via Additive Quantization](https://mlsanity.com/paper/2401.06118/) (2024)
- [SliM-LLM: Salience-Driven Mixed-Precision Quantization for Large Language Models](https://mlsanity.com/paper/2405.14917/) (2024)
- [LQER: Low-Rank Quantization Error Reconstruction for LLMs](https://mlsanity.com/paper/2402.02446/) (2024)
- [Mixture Compressor for Mixture-of-Experts LLMs Gains More](https://mlsanity.com/paper/2410.06270/) (2024)
- [Pushing the Limits of Large Language Model Quantization via the Linearity Theorem](https://mlsanity.com/paper/2411.17525/) (2024)
- [EasyQuant: An Efficient Data-free Quantization Algorithm for LLMs](https://mlsanity.com/paper/2403.02775/) (2024)
- [KurTail : Kurtosis-based LLM Quantization](https://mlsanity.com/paper/2503.01483/) (2025)
- [Cherry on Top: Parameter Heterogeneity and Quantization in Large Language Models](https://mlsanity.com/paper/2404.02837/) (2024)
- [AlphaQ: Calibration-Free Bit Allocation for Mixture-of-Experts Quantization](https://mlsanity.com/paper/2606.04980/) (2026)
- [Beyond Outliers: A Data-Free Layer-wise Mixed-Precision Quantization Approach Driven by Numerical and Structural Dual-Sensitivity](https://mlsanity.com/paper/2603.17354/) (2026)
- [Q-Strata: Hierarchical Bit Allocation for Mixed-Precision Quantization of Mixture-of-Experts LLMs](https://mlsanity.com/paper/2608.30564/) (2026)
- [MixQuant: Adaptive Mixed-Precision Quantization for Large Language Models](https://mlsanity.com/paper/2607.23047/) (2026)
- [GAMMA: Global Bit Allocation for Mixed-Precision Models under Arbitrary Budgets](https://mlsanity.com/paper/2605.18475/) (2026)
- [Fidelity Is Not Safety: Gently-Compressed LLMs Pass Every Data-Free Quality Guard Yet Invent Procedure Steps in Agentic Execution](https://mlsanity.com/paper/2607.28196/) (2026)
- [Beyond Scalar Sensitivity: Activation-Aware Mixed-Precision LLM Quantization with Cross-Layer Refinement](https://mlsanity.com/paper/2609.25916/) (2026)

## Abstract (arXiv)

A single random Gaussian probe gives an unbiased estimate of the squared Frobenius norm of a layer's quantization error. The estimator is well-behaved because round-to-nearest error is spectrally flat. Across 1,683 tensors from a 35B MoE and a 9B dense model, effective dimensionality is 0.93 to 0.96 times the i.i.d. noise value of the same shape, and on the MoE the median is unchanged from 2-bit to 8-bit. The probe coefficient of variation is predictable from tensor shape. One probe measures per-tensor sensitivity to within 4 to 7%; twenty probes reach 1.3 to 1.4%.RAM applies the propagated form of this estimator to budget-targeted mixed-precision quantization with no calibration data. Gaussian probes carrying the network's own input statistics score every tensor at six bit-widths. A knapsack solver allocates bits under an exact byte budget, with guardrails against catastrophic 2-bit assignments. One probe pass serves any budget. Isolated and propagated scores rank tensors independently on Qwen3.5-35B-A3B (Spearman -0.01), yet the propagated probe rank-correlates 0.81 to 0.83 with the GPTQ layer objective from real activations, while the isolated estimator is uncorrelated with it. That objective is the wrong allocation target: at matched bytes on Qwen3.8-27B, a block-output probe beats a vendor IQ3_M mix and an oracle that allocates from the real-activation objective. On Qwen3-8B the propagated probe ties HAWQ-V2 at matched bytes. Across seven architectures from 8B to 122B, with probe timing up to a 400B model in nine minutes on one workstation, RAM reaches 3.5 to 13.6% lower median WikiText-2 perplexity than size-comparable uniform 4-bit builds on the tested MoE models. (Black Sheep Ai baa.ai)