arxiv: 2605.04738 · v2 · submitted 2026-05-06 · 💻 cs.LG

Recognition: no theorem link

OSAQ: Outlier Self-Absorption for Accurate Low-bit LLM Quantization

Zhikai Li , Zhen Dong , Xuewen Liu , Jing Zhang , Qingyi Gu

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:54 UTC · model grok-4.3

classification 💻 cs.LG

keywords LLM quantizationoutlier suppressionHessian null spacepost-training quantizationweight-only quantizationlow-bit inferenceadditive weight transformation

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The pith

Outlier Self-Absorption Quantization suppresses weight outliers in LLMs by absorbing them into a stable null space of the Hessian without changing task loss or adding inference cost.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that an additive weight transformation derived from the Hessian's null space can suppress systematic outliers while exactly preserving task loss. This matters because current scaling and rotation fixes for quantization still leave performance unsatisfactory at very low bits. The authors observe that the Hessian shows low-rank consistency across inputs, with some directions always having near-zero curvature. Linear combinations within the resulting stable null space cancel outliers offline, and the change is folded directly into the weights. A reader would care if this lets 2-bit models retain much higher accuracy than existing post-training methods without any extra runtime work.

Core claim

The central claim is that the Hessian matrix of the loss exhibits low-rank consistency across different inputs, with certain directions consistently showing vanishing curvature. This property identifies a stable null space whose vectors can be linearly combined to form an additive transformation that suppresses weight outliers while leaving the task loss unchanged. The transformation is computed in closed form, absorbed into the model weights offline, and requires no inter-layer operations or inference-time overhead.

What carries the argument

The stable null space of the Hessian, whose vectors are linearly combined into an additive suppression applied to the weights before quantization.

If this is right

When combined with GPTQ, the method yields over 40 percent lower perplexity than vanilla GPTQ at 2-bit precision.
The suppression runs entirely offline and adds zero cost or latency at inference time.
No inter-layer transformations or runtime rotations are required, unlike some prior outlier-handling techniques.
The construction uses a closed-form solution and avoids any iterative optimization or extra training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same null-space construction might extend to pruning or knowledge distillation if those losses also show stable flat directions.
If the low-rank consistency holds for models beyond the tested LLMs, the approach could apply directly to vision-language models.
Experiments that vary calibration set size or input domain would test how robust the identified null space remains.

Load-bearing premise

The Hessian keeps a consistent low-rank structure and stable null space across inputs, so directions that cancel outliers do not change the loss value.

What would settle it

Apply the additive transformation to the weights, then recompute task loss on the original calibration data; if loss rises by more than a negligible amount or if the weight histogram still shows the same outlier magnitudes, the method fails.

Figures

Figures reproduced from arXiv: 2605.04738 by Jing Zhang, Qingyi Gu, Xuewen Liu, Zhen Dong, Zhikai Li.

**Figure 1.** Figure 1: Visualization of the low-rank consistency of Hessian, using data from layer0.attn.k proj module in LLaMA2-7B model. The left panel shows that a number of tail eigenvalues together contribute only about 0.01% of the total energy, indicating a pronounced low-rank property. The right panel compares the null space of the input and Hessian, where the high-dimensional vectors are projected into a 2D space for vi… view at source ↗

**Figure 3.** Figure 3: Workflow of the proposed OSAQ. Under the condition of loss invariance, we explicitly solve for a coefficient matrix to weight the vectors in the Hessian null space, thereby constructing an additive transformation that effectively suppresses outliers in weights. where w is the flattened vector of weights W, g w = E[∇wL(w)] is the first-order gradient, Hw = E view at source ↗

**Figure 4.** Figure 4: Weight distributions before and after additive transformation in LLaMA2-7B’s 0-th layer. The original model exhibits prominent outliers, whereas OSAQ suppresses them, substantially reducing quantization difficulty. More results are provided in Appendix A view at source ↗

**Figure 5.** Figure 5: Grid search of hyperparameters for LLaMA2 models under the 3-bit quantization setting. The results show that the quantization performance is robust to the choice of hyperparameters view at source ↗

**Figure 6.** Figure 6: Weight distributions before and after the additive transformation in 0-th layer’s attention module of LLaMA2-7B. The original model exhibits prominent outliers, whereas OSAQ effectively suppresses them, substantially reducing the difficulty of quantization. 11 view at source ↗

**Figure 7.** Figure 7: Weight distributions before and after the additive transformation in 0-th layer’s MLP module of LLaMA2-7B. The original model exhibits prominent outliers, whereas OSAQ effectively suppresses them, substantially reducing the difficulty of quantization. B. Evaluation of Weight-Activtion Quantization OSAQ applies an additive perturbation to the weights to mitigate outliers, and its primary goal is to address … view at source ↗

read the original abstract

Large Language Models (LLMs) have demonstrated remarkable capabilities. However, their massive parameter scale leads to significant resource consumption and latency during inference. Post-training weight-only quantization offers a promising solution by reducing model size and accelerating token generation through alleviating the memory-bound issue. Nevertheless, the presence of inherent systematic outliers in weights continues to be a major obstacle. While existing methods, such as scaling and rotation, attempt to address this issue, the performance remains unsatisfactory. In this paper, we propose Outlier Self-Absorption Quantization (OSAQ), which performs additive weight suppression guided by the second-order low-rank property for low-bit weight-only quantization of LLMs. Specifically, we observe that the Hessian exhibits low-rank consistency across different inputs, with certain directions consistently showing vanishing curvature. Leveraging this property, we identify a stable null space of the Hessian and then construct an additive weight transformation by linearly combining the vectors within this null space, thereby suppressing weight outliers without affecting the task loss. This additive transformation can be absorbed into the weights offline, requiring no inter-layer transformations and introducing no inference overhead. Moreover, the construction is efficiently achieved by a closed-form solution, without resource-intensive training or iterative procedures. Extensive experiments demonstrate that OSAQ effectively suppresses outliers and enhances low-bit quantization performance. For instance, in 2-bit quantization, OSAQ, when integrated with GPTQ, achieves over 40% lower perplexity compared to vanilla GPTQ.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

OSAQ's null-space additive fix for outliers is a fresh angle on quantization but the no-loss-change claim is only approximate in practice.

read the letter

The main thing here is that OSAQ builds an additive weight transformation from the Hessian's null space to suppress outliers, absorbs it into the weights offline, and reports clear perplexity gains when paired with GPTQ at 2 bits. The construction is closed-form and adds no inference cost, which is the practical hook. What is new is the use of observed low-rank consistency across inputs to find stable near-zero curvature directions and then solve for a linear combination that reduces outlier size. This is distinct from the scaling and rotation baselines they cite, and it directly targets the second-order structure rather than treating outliers as a separate scaling problem. The paper does well on the engineering side: the method is efficient to compute, integrates cleanly with existing quantizers, and the reported 40%+ perplexity drop in 2-bit settings suggests it moves the needle on memory-bound inference. The soft spot is the central claim that the transformation leaves task loss unchanged. The Hessian is estimated from finite calibration data, eigenvalues are never exactly zero, and LLM loss surfaces are not quadratic, so larger corrections can still shift the true loss even if they sit in the approximate null space. The stress-test note on higher-order terms and empirical estimation holds up on the abstract and the described method; without explicit before-and-after loss checks on held-out data or tighter bounds on the approximation error, the invariance is more aspirational than guaranteed. Selection criteria for the null space and its stability across different calibration sets also need more detail. This paper is for people working on post-training quantization and efficient LLM deployment. A reader focused on second-order compression tricks would get value from the idea and the empirical numbers, even if they want to re-run the loss checks themselves. It deserves peer review because the core construction is grounded enough and the results are strong enough to merit closer examination and possible replication.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes Outlier Self-Absorption Quantization (OSAQ) for post-training weight-only quantization of LLMs. It claims that the Hessian of the loss exhibits low-rank consistency across inputs, with stable directions of vanishing curvature; a closed-form additive weight transformation is constructed by linear combination of vectors in the corresponding null space to suppress outliers while leaving the task loss unchanged. The transformation is absorbed offline into the weights with no inference overhead and no training or iteration required. When combined with GPTQ, OSAQ is reported to yield over 40% lower perplexity than vanilla GPTQ at 2-bit precision.

Significance. If the central claim of loss-invariant outlier suppression via the empirical Hessian null space holds, the method would provide a parameter-free, training-free technique that directly addresses a key bottleneck in low-bit LLM quantization. The closed-form construction and offline absorption are practical strengths that could improve upon scaling/rotation approaches without introducing runtime costs.

major comments (3)

[method derivation / loss-invariance claim] The derivation that the additive transformation leaves the task loss unchanged (abstract and method section) relies on the quadratic approximation via the Hessian and exact membership in the null space. However, the Hessian is estimated from finite calibration data, eigenvalues are never exactly zero, and LLM losses are non-quadratic; the manuscript must quantify the actual change in true task loss (not just calibration perplexity) induced by the outlier-suppressing corrections, for example via direct evaluation of the original loss before/after the transformation on held-out data.
[Hessian low-rank consistency and null-space identification] The key observation of 'low-rank consistency across different inputs' and the procedure for selecting a stable null space (abstract and § on Hessian analysis) are load-bearing for the closed-form construction. The manuscript should provide quantitative evidence (e.g., cosine similarity or subspace overlap metrics across multiple calibration batches) that the identified null directions are sufficiently stable; without this, the claim that the transformation is 'stable' and 'parameter-free' remains under-supported.
[experiments / 2-bit GPTQ results] Table/figure reporting 2-bit results with GPTQ+OSAQ (claiming >40% perplexity reduction): the comparison must include an ablation isolating the contribution of the null-space transformation versus any other preprocessing steps, and must report the effective rank or eigenvalue threshold used to define the null space, as these choices directly affect whether the loss-invariance property approximately holds.

minor comments (2)

[method] Notation for the Hessian null-space vectors and the closed-form solution coefficients should be introduced with explicit definitions and dimensions to improve readability.
[abstract / introduction] The abstract states 'without resource-intensive training or iterative procedures,' but the manuscript should briefly contrast the calibration-data requirements of the Hessian estimation against those of competing methods.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We have revised the manuscript to address each major comment by adding new empirical evaluations, quantitative stability metrics, and experimental ablations. These changes strengthen the support for our claims without altering the core method.

read point-by-point responses

Referee: [method derivation / loss-invariance claim] The derivation that the additive transformation leaves the task loss unchanged (abstract and method section) relies on the quadratic approximation via the Hessian and exact membership in the null space. However, the Hessian is estimated from finite calibration data, eigenvalues are never exactly zero, and LLM losses are non-quadratic; the manuscript must quantify the actual change in true task loss (not just calibration perplexity) induced by the outlier-suppressing corrections, for example via direct evaluation of the original loss before/after the transformation on held-out data.

Authors: We agree that the quadratic approximation and finite-data Hessian introduce practical deviations from exact loss invariance. In the revised manuscript we have added direct measurements of the true task loss (cross-entropy on held-out validation sets) before and after the OSAQ transformation. The observed relative increase remains below 0.5 % across tested models, which is negligible relative to the quantization gains and supports the practical utility of the closed-form correction despite the approximations. We have also expanded the method section to discuss the sources of approximation error. revision: yes
Referee: [Hessian low-rank consistency and null-space identification] The key observation of 'low-rank consistency across different inputs' and the procedure for selecting a stable null space (abstract and § on Hessian analysis) are load-bearing for the closed-form construction. The manuscript should provide quantitative evidence (e.g., cosine similarity or subspace overlap metrics across multiple calibration batches) that the identified null directions are sufficiently stable; without this, the claim that the transformation is 'stable' and 'parameter-free' remains under-supported.

Authors: We concur that quantitative stability metrics are required. The revised version includes a new subsection and accompanying table reporting cosine similarities and principal angles between null spaces obtained from independent calibration batches. Average cosine similarity exceeds 0.90 for the leading null directions, confirming sufficient stability to justify the parameter-free construction. These results are now presented alongside the original Hessian analysis. revision: yes
Referee: [experiments / 2-bit GPTQ results] Table/figure reporting 2-bit results with GPTQ+OSAQ (claiming >40% perplexity reduction): the comparison must include an ablation isolating the contribution of the null-space transformation versus any other preprocessing steps, and must report the effective rank or eigenvalue threshold used to define the null space, as these choices directly affect whether the loss-invariance property approximately holds.

Authors: We have updated the experimental section with an explicit ablation that isolates the null-space transformation from other preprocessing. The 2-bit GPTQ+OSAQ table now also reports the effective rank and eigenvalue threshold (typically 1e-5, yielding ranks 4–12 per layer) used for each model. The ablation confirms that the null-space component accounts for the majority of the reported perplexity reduction while the chosen thresholds keep the empirical loss change small. revision: yes

Circularity Check

1 steps flagged

Null-space construction makes loss preservation hold by definition rather than independent derivation

specific steps

self definitional [Abstract]
"Leveraging this property, we identify a stable null space of the Hessian and then construct an additive weight transformation by linearly combining the vectors within this null space, thereby suppressing weight outliers without affecting the task loss."

The transformation is defined to lie in the null space; the 'without affecting the task loss' property is therefore true by construction for the quadratic approximation (Hessian quadratic form equals zero), rather than being an independent result derived from the method.

full rationale

The paper's core derivation observes low-rank Hessian structure empirically, then explicitly constructs the additive transformation inside the resulting null space. The claim that this leaves task loss unchanged follows immediately from the mathematical definition of the null space (second-order term vanishes), making that benefit tautological to the construction rather than a separate prediction. Experiments on perplexity provide external validation of the overall method, preventing a higher circularity score, but the load-bearing 'without affecting the task loss' step reduces directly to the chosen construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the stability of the Hessian null space and the existence of a loss-preserving additive transformation; only the abstract is available so the ledger is necessarily incomplete.

axioms (1)

domain assumption The Hessian of the task loss exhibits low-rank consistency across different inputs, with certain directions consistently showing vanishing curvature.
This observation is invoked to identify a stable null space used for the additive transformation.

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