# GPTQ-2D: Cubic-Time Two-Sided Adaptive Rounding

arXiv: https://arxiv.org/abs/2607.27042 · HTML: https://mlsanity.com/paper/2607.27042/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 9 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning](https://mlsanity.com/paper/2208.11580/) (2022)
- [Model-Preserving Adaptive Rounding](https://mlsanity.com/paper/2505.22988/) (2025)
- [The Geometry of LLM Quantization: GPTQ as Babai's Nearest Plane Algorithm](https://mlsanity.com/paper/2507.18553/) (2025)
- [The Lattice Geometry of Neural Network Quantization -- A Short Equivalence Proof of GPTQ and Babai's Algorithm](https://mlsanity.com/paper/2508.01077/) (2025)

## Built on by (most cited first, 2 of 2)

- [FastKron: Efficient Quantization with Kronecker-Factored Hessians](https://mlsanity.com/paper/2608.06291/) (2026)
- [WaterKron and FlipFlop Hessian: Information-Theoretically Grounded Quantization with Kronecker-factored Hessians](https://mlsanity.com/paper/2609.14706/) (2026)

## Abstract (arXiv)

Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.