The Geometry of Ignorance: Detecting Suppressed Knowledge in LLMs through Activation Fragmentation
Organizations: Technion - Israel Institute of Technology
Abstract
Evaluating whether a large language model (LLM) genuinely knows a concept remains a fundamental problem, since behavioral metrics can be fooled by suppression: a model that refuses to answer may still harbor recoverable knowledge. The framework measures the geometric coherence of anon-vectors, the differences in hidden states between a concept-specific query and its anonymized paraphrase, and shows that genuine knowledge forms a tight, low-entropy cluster while ignorance forces geometric fragmentation. Across ten instruction-tuned LLMs from three model families, all fragmentation metrics reliably separate known from unknown concepts, and post-unlearning, fragmentation geometry correlates with adversarial recovery across four attack types and five unlearning algorithms, though the direction of that relationship is shown to depend on attack methodology. Furthermore, a layer-resolved analysis reveals that different adversarial mechanisms exploit this geometry at varying depths, with high attack-specific correlations that are invisible to behavioral evaluation. Together, these results constitute a single-phase, attack-free proxy for unlearning robustness that captures aspects of parametric knowledge inaccessible to output-level metrics. A reference implementation is available at https://github.com/AdarPeleg/geometry-of-ignorance
Figures & tables
| Metric | Gap [95% CI] | ||
| Dispersion ( HP expected; 10/10 models) | |||
| AUSS | |||
| AUSS | |||
| NormAnonDiff | |||
| AnonDiff | |||
| Reg | |||
| Metric | ICL | GCG | Steering |
| AUSS | – | – | |
| AUSS | – | – | |
| – | – | ||
| Reg | – | – | |
| Reg | – | – | |
| AnonDiff | – | – |
Appendix figures & tables14 assets
Supplementary material from the paper’s appendix.
Appendix
| Metric | ( ) | |
| / Global_Coh. | 2 | 0.14 (.82) |
| 5 | 0.64 (.13) | |
| 10 | 0.72 (.13) | |
| 20 | 0.92 (.04) | |
| 50 | 1.13 (.02) | |
| Batch_SecondMoment_TopEig | 2 | 0.19 (.73) |
| Metric | ||
| 0.2 | 0.42 | |
| 0.3 | 0.67 | |
| 0.4 | 1.34 | |
| 0.5 | 1.51 | |
| 0.6 | 1.72 | |
| 0.7 | 2.21 |
| Harry Potter (known) | TOFU (unknown, QA = 0.00) | ||||||
| Model | QA | AUSS | AUSS | AUSS | AUSS | ||
| Qwen family | |||||||
| Qwen-7B-Chat | 0.083 | 0.752 | 0.952 | 0.577 | 0.639 | 1.309 | 0.683 |
| Qwen2.5-7B | 0.250 | 0.684 | 1.150 | 0.501 | 0.601 | 1.402 | 0.408 |
| Qwen-14B-Chat | 0.135 | 0.804 | 0.780 | 0.553 | 0.688 | 1.164 | 0.696 |
| Qwen2.5-14B | 0.323 | 0.697 | 1.116 | 0.522 | 0.689 | 1.151 | 0.440 |
| Harry Potter (known) | TOFU (unknown, QA = 0.00) | |||||||||
| Reg | AnonDiff | Reg | AnonDiff | |||||||
| Model | ||||||||||
| Qwen family | ||||||||||
| Qwen-7B-Chat | 0.007 | 0.995 | 3.808 | 20.4 | 0.138 | 0.043 | 0.969 | 4.769 | 16.9 | 0.098 |
| Qwen2.5-7B | 0.002 | 0.999 | 3.715 | 29.5 | 0.118 | 0.009 | 0.996 | 4.614 | 24.7 | 0.121 |
| Qwen-14B-Chat | 0.018 | 0.990 | 3.544 | 21.5 | 0.166 | 0.090 | 0.954 | 4.795 | 16.7 | 0.088 |
| Metric | vs. TOFU | ( ) |
| / Global_Coh. | HP-forget | (.0039) |
| HP-retain | (.0039) | |
| Star Wars | (.0039) | |
| Batch_SecondMoment_TopEig | HP-forget | (.0039) |
| HP-retain | (.0039) | |
| Star Wars | (.0039) |
| Logistic Regr. | MLP (8 hidden) | |
| Accuracy | 100% | 94.4% |
| AUC | 1.00 | 1.00 |
| Correct | 18/18 | 17/18 |
| Models fully correct | 9/9 | 8/9 |
| Metric | Steering | ICL | GCG | MIA |
| AUSS dispersion | ||||
| AUSS | ||||
| AUSS | ||||
| AnonDiff | ||||
| AnonDiff | ||||
| Family | Metric | Attack | |||
| AUSS | Centroid_Norm | ICL | +0.551 | 0.00002 | 0.0001 ** |
| AUSS | AUSS | ICL | 0.572 | 0.00001 | 0.0001 ** |
| AUSS | AUSS L2 | ICL | 0.551 | 0.00002 | 0.0001 ** |
| AUSS | Global_Coherence | ICL | +0.551 | 0.00002 | 0.0001 ** |
| AUSS | Batch_Cov_Trace | ICL | 0.551 | 0.00002 | 0.0001 ** |
| AUSS | Batch_SecMom_TopEig | ICL | +0.557 | 0.00001 | 0.0001 ** |
| Attack | Family | Best layer | Best metric | / | ||
| GCG | AUSS | L29 | AUSS | / | ||
| Entropy | L30 | NormAnonDiff | / | |||
| ICL | AUSS | L28 | Batch_Cov_TopEig | / | ||
| Entropy | L00 | AnonDiff | / | |||
| MIA | AUSS | L31 | Centroid_Norm | / | ||
| Entropy | L32 | Reg | / |
| Attack | Metrics with 0 LOO sign flips | BH-significant ( ) |
| ICL | 8/8 | 7/8 |
| Steering | 8/8 | 0/8 |
| GCG | 7/8 | 0/8 |
| MIA | 2/8 | 0/8 |
| Forget | Geometry | Attack Success | |||||||||
| Method | Cpt | ROUGE-L | AUSS | AUSS | Steer-AUC | Steer-WF | ICL | MIA | GCG | ||
| Base | HP | 0.131 | 0.878 | 0.507 | 0.439 | 0.428 | 0.265 | +0.000 | 0.827 | 0.000 | |
| SW | 0.152 | 0.876 | 0.516 | 0.447 | 0.449 | 0.296 | +0.120 | 0.955 | 0.000 | ||
| WS | 0.090 | 0.911 | 0.379 | 0.341 | 0.503 | 0.274 | +0.040 | 0.837 | 0.000 | ||
| GA | HP | ✓ | 0.000 | 0.823 | 0.717 | 0.553 | 0.500 | 0.000 | +0.000 | 0.678 | 0.000 |
| SW | ✓ | 0.000 | 0.875 | 0.521 | 0.442 | 0.500 | 0.000 | +0.000 | 0.695 | 0.000 | |
| Forget | Geometry | Attack Success | |||||||||
| Method | Cpt | ROUGE-L | AUSS | AUSS | Steer-AUC | Steer-WF | ICL | MIA | GCG | ||
| Base | HP | 0.062 | 0.856 | 0.594 | 0.493 | 0.423 | 0.437 | +0.000 | 0.915 | 0.000 | |
| SW | 0.113 | 0.873 | 0.529 | 0.455 | 0.424 | 0.379 | +0.040 | 0.987 | 0.000 | ||
| WS | 0.068 | 0.901 | 0.416 | 0.370 | 0.425 | 0.347 | +0.040 | 0.952 | 0.000 | ||
| GA | HP | ✓ | 0.000 | 0.872 | 0.534 | 0.439 | 0.500 | 0.000 | +0.000 | 0.700 | 0.000 |
| SW | ✓ | 0.000 | 0.801 | 0.797 | 0.591 | 0.500 | 0.000 | +0.000 | 0.102 | 0.000 | |
| Forget | Geometry | Attack Success | |||||||||
| Method | Cpt | ROUGE-L | AUSS | AUSS | Steer-AUC | Steer-WF | ICL | MIA | GCG | ||
| Base | HP | 0.106 | 0.872 | 0.534 | 0.439 | 0.433 | 0.373 | +0.040 | 0.982 | 0.000 | |
| SW | 0.098 | 0.866 | 0.557 | 0.473 | 0.482 | 0.261 | +0.080 | 0.982 | 0.000 | ||
| WS | 0.080 | 0.907 | 0.395 | 0.352 | 0.495 | 0.373 | +0.000 | 0.938 | 0.000 | ||
| GA | HP | ✓ | 0.000 | 0.936 | 0.275 | 0.253 | 0.500 | 0.000 | +0.000 | 0.119 | 0.000 |
| SW | ✓ | 0.000 | 0.901 | 0.417 | 0.370 | 0.500 | 0.000 | +0.000 | 0.461 | 0.000 | |
Factual Knowledge in Language Models
Factual knowledge in language models concerns what facts they store, how reliably they retrieve them, and how that knowledge can be tested or changed.
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