Norm-Preserving Biprojected Abliteration

Abliteration is a technique for removing refusal behaviors from language models by identifying and intervening on "refusal directions" in activation space, notionally represented via a single mean refusal direction. This finding has been useful in mechanistic interpretability.

We recently presented a refinement called "projected abliteration" that improves upon the conventional approach by removing only the mechanistically relevant components of the refusal direction, confirming a prior finding that LLMs encode refusal and harmfulness separately. After that, we further refined the technique to "biprojected abliteration", which also removed the corresponding component when removing refusal measured using one layer from another layer entirely; in principle this would avoid disturbing the harmless direction of any layer targeted for intervention. Interestingly, some safety refusal came back.

Upon further consideration, there remained an issue with regard to layer weight modification and weight norms.

In conventional (and our previously modified) abliteration, the normalized refusal direction is subtracted from the layer's residual streams targeted for intervention, specifically self_attn.o_proj and mlp.down_proj. Although this is effective in practice for steering, it is mathematically unprincipled because:

• the direction removed contains a unit magnitude component in addition to the directional component, complicating interpretation,
• the removal does not respect the relative importance of neurons, resulting in unpredictable scale effects, and
• disturbs the geometry of the weight matrix in unpredictable ways.

Contrary to conventional wisdom that abliteration significantly degrades model capabilities, our norm-preserving approach improved reasoning performance over the baseline model (NatInt: 21.33 vs 18.72), while achieving effective refusal removal (UGI: 32.61 vs 19.58).

A Refined Mathematical Intervention

Instead of subtracting the refusal direction from the target weights, we propose subtracting only the directional component while preserving the norm of the weights.

Applying norm-preservation is more respectful of existing layer normalization by maintaining the relative activation scale structure that the model's normalization layers were trained to expect. We should therefore expect some improvement over naive abliteration with regard to reducing incidental damage to reasoning. Furthermore, the ablation can still be performed as a rank-1 modification, keeping the overall approach computationally efficient.

Given:

• Weight matrix $\mathbf{W} \in \mathbb{R}^{d_{\text{out}} \times d_{\text{in}}}$
• Refusal direction $\mathbf{r} \in \mathbb{R}^{d_{\text{out}}}$ (refined via biprojection)
• Scaling factor $\alpha \in [0, 1]$

Step 1: Normalize the refusal direction

$$\widehat{\mathbf{r}}=\frac{\mathbf{r}}{\mathrm{\parallel }\mathbf{r}{\mathrm{\parallel }}_2}\backslash hat\{\backslash mathbf\{r\}\}\; =\; \backslash frac\{\backslash mathbf\{r\}\}\{\backslash |\backslash mathbf\{r\}\backslash |\_2\}$$

Step 2: Decompose weight matrix into magnitude and direction

For each row $i$ of $\mathbf{W}$:

$$m_i=\mathrm{\parallel }{\mathbf{W}}_{i,:}{\mathrm{\parallel }}_{2}m\_i\; =\; \backslash |\backslash mathbf\{W\}\_\{i,:\}\backslash |\_2$$

$${\widehat{\mathbf{W}}}_{i,:}=\frac{{\mathbf{W}}_{i,:}}{\mathrm{\parallel }{\mathbf{W}}_{i,:}{\mathrm{\parallel }}_2}\backslash hat\{\backslash mathbf\{W\}\}\_\{i,:\}\; =\; \backslash frac\{\backslash mathbf\{W\}\_\{i,:\}\}\{\backslash |\backslash mathbf\{W\}\_\{i,:\}\backslash |\_2\}$$

Or in matrix form: $$\mathbf{M}=\text{diag}(\mathrm{\parallel }{\mathbf{W}}_{1,:}{\mathrm{\parallel }}_2,\dots ,\mathrm{\parallel }{\mathbf{W}}_{d_{\text{out}},:}{\mathrm{\parallel }}_2)\backslash mathbf\{M\}\; =\; \backslash text\{diag\}(\backslash |\backslash mathbf\{W\}\_\{1,:\}\backslash |\_2,\; \backslash ldots,\; \backslash |\backslash mathbf\{W\}\_\{d\_\{\backslash text\{out\}\},:\}\backslash |\_2)$$

$$\widehat{\mathbf{W}}={\mathbf{M}}^{-1}\mathbf{W}\backslash hat\{\backslash mathbf\{W\}\}\; =\; \backslash mathbf\{M\}^\{-1\}\backslash mathbf\{W\}$$

Step 3: Ablate refusal direction from the normalized directional component

Compute projection coefficients (alignment of each input dimension with refusal): $$\mathbf{p}={\widehat{\mathbf{r}}}^T\widehat{\mathbf{W}}\in {\mathbb{R}}^{d_{\text{in}}}\backslash mathbf\{p\}\; =\; \backslash hat\{\backslash mathbf\{r\}\}^T\; \backslash hat\{\backslash mathbf\{W\}\}\; \backslash in\; \backslash mathbb\{R\}^\{d\_\{\backslash text\{in\}\}\}$$

Remove the refusal component via rank-1 update: $${\widehat{\mathbf{W}}}_{\text{ablated}}=\widehat{\mathbf{W}}-\alpha \cdot \widehat{\mathbf{r}}{\mathbf{p}}^T\backslash hat\{\backslash mathbf\{W\}\}\_\{\backslash text\{ablated\}\}\; =\; \backslash hat\{\backslash mathbf\{W\}\}\; -\; \backslash alpha\; \backslash cdot\; \backslash hat\{\backslash mathbf\{r\}\}\; \backslash mathbf\{p\}^T$$

Renormalize each row to unit length, to enable recombination with original magnitudes: $${\widehat{\mathbf{W}}}_{\text{new}}=\text{normalize}({\widehat{\mathbf{W}}}_{\text{ablated}},\text{dim}=1)\backslash hat\{\backslash mathbf\{W\}\}\_\{\backslash text\{new\}\}\; =\; \backslash text\{normalize\}(\backslash hat\{\backslash mathbf\{W\}\}\_\{\backslash text\{ablated\}\},\; \backslash text\{dim\}=1)$$

Step 4: Recombine with original magnitudes

$${\mathbf{W}}_{\text{new}}=\mathbf{M}{\widehat{\mathbf{W}}}_{\text{new}}\backslash mathbf\{W\}\_\{\backslash text\{new\}\}\; =\; \backslash mathbf\{M\}\; \backslash hat\{\backslash mathbf\{W\}\}\_\{\backslash text\{new\}\}$$

This ensures that $|\mathbf{W}_{\text{new}, i,:}|2 = |\mathbf{W}{i,:}|_2$ for all rows $i$, preserving the learned importance structure while redirecting computation away from the refusal direction.

Sample PyTorch implementation

import torch

# Core implementation (excerpt from complete function)
"""
    Args:
        W: Weight matrix of shape [out_features, in_features]
        refusal_dir: Refusal direction vector of shape [out_features]
        scale_factor: Scaling factor for ablation strength (default: 1.0)
"""

        # Normalize refusal direction
        refusal_normalized = torch.nn.functional.normalize(refusal_dir, dim=0)

        # Decompose weight matrix
        W_norm = torch.norm(W, dim=1, keepdim=True)  # [out_features, 1]
        W_direction = torch.nn.functional.normalize(W, dim=1)  # normalized per output neuron
    
        # Apply abliteration to the DIRECTIONAL component
        projection = torch.matmul(refusal_normalized, W_direction)  # [in_features]
        W_direction_new = W_direction - scale_factor * torch.outer(refusal_normalized, projection)
    
        # Re-normalize the adjusted direction to enable recombination
        W_direction_new = torch.nn.functional.normalize(W_direction_new, dim=1)
    
        # Recombine: keep original magnitude, use new direction
        W_new = W_norm * W_direction_new

Layer Selection Methodology

We measured refusal directions across all layers but required a principled approach to select which measurements to use for intervention. Our selection employed a composite quality metric combining three factors:

Signal-to-noise ratio (SNR): The magnitude of the refusal direction relative to the mean activations:

snr = ||r|| / max(||harmful_mean||, ||harmless_mean||)

where the refusal direction $\mathbf{r} = \text{harmful_mean} - \text{harmless_mean}$.

Cosine dissimilarity: The angular separation between harmful and harmless activation means:

dissimilarity = 1 - cosine_similarity(harmful_mean, harmless_mean)

Higher values indicate more distinct representational geometry.

Composite quality score:

quality = snr × (1 - cos_sim)

By charting these metrics across all layers, we selected candidates exhibiting both high SNR and strong cosine dissimilarity, with particular attention to layers showing sharp changes in these metrics. The suitability of a selected refusal direction for application to nearby and preceding layers was informed by tracking the cosine similarity evolution of refusal directions across consecutive layers—stable directional alignment suggested robust cross-layer applicability.

This approach is admittedly heuristic, but grounded in the observed geometric structure of refusal representations. Future work might formalize optimal layer selection through systematic ablation studies.

This heuristic is computationally efficient: a single inference pass captures activations across all layers, and quality metrics are computed post-hoc from this data. Unlike iterative search approaches that require multiple model evaluations, the analysis adds negligible overhead to the standard abliteration workflow, merely extracting more signal from measurements already performed.

For Gemma3 12B Instruct, with layers numbered [0..47], we picked the measurements from layers 23 and 29 for broad application. Keeping the refusal and mean harmless direction measurements proved to be vital in subsequent refinements

Result

With this revised method, we abliterated Gemma3 12B Instruct yet again. As before, we applied a default scale factor of 1.0, intervening on layers [11..41]. As expected, we were able to bypass refusal with harmful test prompts. The model retained more of its capabilities in informal testing, and "grimjim/gemma-3-12b-it-norm-preserved-biprojected-abliterated" scored highest on the UGI and NatInt benchmarks on the UGI Leaderboard compared to our prior published abliteration variants for the same baseline model, and the baseline Instruct model itself.

As before, during measurement of activations, magnitude sparsification at 0.995 strength was applied when obtaining measurements from prompts. This was necessary to distinguish the refusal direction between the mean harmful and harmless directions. The empirical observation was that strong outlier activations characterize the model.

To maximize numerical stability, we continued to employ 32-bit floating point throughout for intermediate calculations even though the model was released in 16-bit bfloat16 floating point format. It was previously noted that performing intermediate calculations in 16-bit bfloat16 led to suboptimal results. We recommend that at least 32-bit floating point be employed in models which evince a large variance in activation magnitudes.

Discussion

By successfully narrowing down the intervention to only the directional component with preserved norms, we establish that refusal direction alone is critical to abliteration outcomes, rather than refusal direction entangled with magnitude effects. However, despite this theoretical grounding, it remains probable that removing even the directional component entangled with harmfulness assessment could reduce safety in unprincipled ways. Korznikov et al. (2025) demonstrated that activation steering on even benign features can compromise LLM safety, suggesting that interventions in representational space may have unintended consequences for safety mechanisms.

Preserving the magnitude was likely important in the case of Gemma3 12B Instruct, given that high magnitude outliers obscured the underlying refusal direction almost certainly encode important behavioral information that should be preserved in order to retain functionality, as reported by Sun et al. (2024).

Benchmark results on the UGI Leaderboard demonstrated clear improvements over prior abliteration variants:

Notably, while standard abliteration achieved comparable uncensoring (UGI scores), it showed slight capability degradation (NatInt: 18.64 vs baseline 18.72). The norm-preserving approach not only matched the uncensoring effectiveness but significantly improved reasoning capability (NatInt: 21.33). This finding aligns with recent observations of a 'Safety Tax' phenomenon (Huang et al., 2025), where safety alignment has been shown to degrade reasoning capabilities in language models. The improvement over baseline suggests that removing directionally-encoded safety constraints may unlock latent reasoning capabilities that were suppressed by safety mechanisms, though this relationship warrants further investigation.

Although we had previously empirically established that intervention on multiple layers was required to achieve the desired rate of compliance to harmful prompts, we found theoretical grounding in a 2023 paper by McGrath et al. titled "The Hydra Effect: Emergent Self-repair in Language Model Computations". The authors demonstrated that when individual layers are ablated, other layers adaptively compensate to restore approximately 70% of the original computation. This self-repair mechanism explains why single-layer interventions generally prove insufficient for robust abliteration, as the model inherently routes around localized damage.

A multi-layer intervention strategy directly addresses this challenge: by simultaneously modifying both attention output projections and MLP down projections across multiple layers, one can effectively "cut multiple heads of the hydra" at the same time, preventing the compensatory mechanisms from restoring refusal behavior. Via judicious selection of layer measurements and a set of intervention layers, a structured rank-2L intervention (where L is the number of targeted layers) can provide sufficient coverage to overcome emergent self-repair while remaining computationally efficient through localized rank-1 updates at each weight matrix. This "hydra effect" in retrospect accounted for the partial return of safety refusals during "biprojected abliteration".

We restructured the traditional abliteration pipeline into three distinct phases: (1) measurement across all layers, (2) analytical layer selection via quality metrics, and (3) targeted intervention on selected layers. This separation provides flexibility: rather than committing to a single 'best' layer, practitioners can select multiple high-quality candidates for intervention, enabling the multi-layer strategy necessary to overcome the hydra effect while maintaining computational efficiency through the rank-1 structure of each individual weight modification.

Finally, an interesting practical consequence emerges from this understanding: the number of intervention layers provides a crude but effective mechanism for regulating the compliance-safety tradeoff. Fewer layers allow more self-repair, preserving some refusal capability, while more layers overcome the compensatory mechanisms more thoroughly. This provides practitioners with a tunable parameter for calibrating model behavior according to their use case and risk tolerance.

References

• Arditi et al, "Refusal in LLMs is mediated by a single direction", lesswrong.com, 2024.
• Huang et al, "Safety Tax: Safety Alignment Makes Your Large Reasoning Models Less Reasonable", arXiv preprint, 2025
• Korznikov et al, "The Rogue Scalpel: Activation Steering Compromises LLM Safety", arXiv preprint, 2025
• Labonne, "Uncensor any LLM with abliteration", huggingface.co, 2024.
• Lai, "Projected Abliteration", HuggingFace article, 2025
• McGrath et al, "The Hydra Effect: Emergent Self-repair in Language Model Computations", arXiv preprint, 2023
• Sun et al,, "Massive Activations in Large Language Models", arXiv preprint, 2024
• Zhao et al, "LLMs Encode Harmfulness and Refusal Separately", arXiv preprint, 2025.

Initial publication date: November 6, 2025.