Preserve-Then-Quantize: Balancing Rank Budgets for Quantization Error Reconstruction in LLMs

Abstract: Quantization Error Reconstruction (QER) reduces accuracy loss in Post-Training Quantization (PTQ) by approximating weights as $\mathbf{W} \approx \mathbf{Q} + \mathbf{L}\mathbf{R}$, using a rank-$r$ correction to reconstruct quantization error. Prior methods devote the full rank budget to error reconstruction, which is suboptimal when $\mathbf{W}$ has intrinsic low-rank structure and quantization corrupts dominant directions. We propose Structured Residual Reconstruction (SRR), a rank-allocation framework that preserves the top-$k$ singular subspace of the activation-scaled weight before quantization, quantizes only the residual, and uses the remaining rank $r-k$ for error reconstruction. We derive a theory-guided criterion for selecting $k$ by balancing quantization-exposed energy and unrecoverable error under rank constraints. We further show that resulting $\mathbf{Q} + \mathbf{L}\mathbf{R}$ parameterization naturally supports Quantized Parameter-Efficient Fine-Tuning (QPEFT), and stabilizes fine-tuning via gradient scaling along preserved directions. Experiments demonstrate consistent perplexity reductions across diverse models and quantization settings in PTQ, along with a 5.9 percentage-point average gain on GLUE under 2-bit QPEFT.