Post-training quantization is widely used for compressing large neural networks, but aggressive low-bit quantization can significantly degrade model quality. A common remedy is to augment the quantized weights with a low-rank correction, leading to approximations of the form $W\approx Q+LR$. In this paper, we study this low-precision plus low-rank representation through the layer-wise reconstruction objective $\|XW-X(Q+LR)\|_F^2$, where $X$ is a calibration matrix. We establish, to our knowledge, the first information-theoretic lower bounds for this problem under finite-alphabet and bounded low-rank compensation constraints. We then propose GPTQ-intrinsic LoRA, a training-free algorithm that incorporates the low-rank correction directly into a GPTQ-style quantization pass by appropriately augmenting the calibration Hessian. For the choice $L=V_r$, where $V_r$ contains the top right singular vectors of $X$, we prove layer-wise reconstruction error bounds in which the usual GPTQ dependence on $\|X\|_F^2$ is replaced by the rank-$r$ residual $\|X-X_r\|_F^2$, up to regularization terms. Under natural structural assumptions, these bounds match the information-theoretic lower bounds in their dominant scaling, up to constants and mild factors. We also introduce Bid-Up, a fixed-grid quantization refinement step that can be alternated with optimal low-rank compensation with guaranteed non-increasing layer-wise reconstruction error. Experiments on Qwen3 language models and DeiT vision transformers show that GPTQ-intrinsic LoRA improves over GPTQ and GPTQ followed by low-rank compensation, with additional gains from refinement loops.

翻译：后训练量化被广泛用于压缩大型神经网络，但激进的低位量化会显著降低模型质量。常见补救措施是将量化权重与低秩修正项结合，得到形如$W\approx Q+LR$的近似表示。本文通过逐层重建目标$\|XW-X(Q+LR)\|_F^2$（其中$X$为校准矩阵）研究这种低精度加低秩表示方法。我们首次在有限字母表和有界低秩补偿约束下，建立了该问题的信息论下界。随后提出GPTQ-本征LoRA——一种免训练算法，通过适当增广校准Hessian矩阵，在GPTQ式量化过程中直接融入低秩修正。针对$L=V_r$（$V_r$包含$X$的顶部右奇异向量）的选取，我们证明逐层重建误差界中，GPTQ通常依赖的$\|X\|_F^2$项被秩$r$残差$\|X-X_r\|_F^2$所替代（正则项不计）。在自然结构假设下，这些误差界的主导尺度（除常数和温和因子外）与信息论下界相匹配。我们还引入Bid-Up——一种固定网格量化细化步骤，可与最优低秩补偿交替执行，保证逐层重建误差非增。在Qwen3语言模型和DeiT视觉Transformer上的实验表明，GPTQ-本征LoRA优于GPTQ及GPTQ后续低秩补偿方法，且通过细化循环可进一步获得性能提升。