# WaterKron and FlipFlop Hessian: Information-Theoretically Grounded Quantization with Kronecker-factored Hessians

arXiv: https://arxiv.org/abs/2609.14706 · HTML: https://mlsanity.com/paper/2609.14706/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 27 references that are in the map.
- First posted on arXiv 13 Sep 2026.

## Builds on (most cited first, 22 of 22)

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- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [QTIP: Quantization with Trellises and Incoherence Processing](https://mlsanity.com/paper/2406.11235/) (2024)
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- [The Lattice Geometry of Neural Network Quantization -- A Short Equivalence Proof of GPTQ and Babai's Algorithm](https://mlsanity.com/paper/2508.01077/) (2025)
- [High-Rate Quantized Matrix Multiplication I](https://mlsanity.com/paper/2601.17187/) (2026)
- [WaterSIC: Information-Theoretically (Near) Optimal Linear Layer Quantization](https://mlsanity.com/paper/2603.04956/) (2026)
- [Core Shrinkage Covariance Estimation for Matrix-variate Data](https://mlsanity.com/paper/2207.12484/) (2022)
- [GPTQ-2D: Cubic-Time Two-Sided Adaptive Rounding](https://mlsanity.com/paper/2607.27042/) (2026)
- [SoftWater: Class-Aware Rate Allocation for Softmax Quantization](https://mlsanity.com/paper/2608.12026/) (2026)
- [FastKron: Efficient Quantization with Kronecker-Factored Hessians](https://mlsanity.com/paper/2608.06291/) (2026)

## Abstract (arXiv)

How should a Kronecker-factored Hessian approximation be chosen for post-training quantization? We address this question through WaterKron, which combines two-sided GPTQ with row- and column-dependent waterfilling scales and entropy coding. We derive its high-rate distortion with respect to the full Hessian using an explicit Kronecker-Hessian mismatch factor Phi. This factor quantifies the asymptotic distortion penalty due to the Kronecker Hessian approximation and provides a criterion for selecting the factors optimally. Minimizing Phi leads to a Gaussian covariance-fitting problem with classical ``flip-flop'' updates. We thus give a rate-distortion justification for using the resulting FlipFlop Hessian in quantization. We evaluate it empirically, finding that the FlipFlop Hessian consistently improves KL divergence and perplexity over input-only, marginal, and Frobenius-based Hessian choices.