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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.14808 |
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| _version_ | 1866915626006609920 |
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| author | von Strauss, Mikael |
| author_facet | von Strauss, Mikael |
| contents | Under real-analytic assumptions on decoder-only Transformers, recent work shows that the map from discrete prompts to last-token hidden states is generically injective on finite prompt sets. We refine this picture: for each layer $\ell$ we define a collision discriminant $Δ^\ell \subset Θ$ and injective stratum $U^\ell = Θ\setminus Δ^\ell$, and prove a dichotomy -- either the model is nowhere injective on the set, or $U^\ell$ is open and dense and every $F^\ell_θ$ is injective. Under mild non-singularity assumptions on the optimizer and an absolutely continuous initialization, generic injectivity persists along smooth training trajectories over any fixed horizon. We also treat symmetry groups $G$, showing that discriminants and injective strata descend to the quotient $Θ/G$, so injectivity is naturally a property of functional equivalence classes.
We complement these results with an empirical study of layerwise geometric diagnostics. We define a separation margin and a co-Lipschitz (lower Lipschitz) constant between prompt space and last-token representation space, estimated via nearest-neighbor statistics on large prompt sets. Applying these diagnostics to pretrained LLaMA-3 and Qwen models, we study behavior across layers, sequence lengths, model scales, and 8- and 4-bit activation quantization. On our sampled prompts we see no collisions in full precision or at 8 bits, while 4-bit quantization induces a small number of collisions and markedly shrinks co-Lipschitz estimates. For a small GPT-2 trained from scratch, normalized metrics remain stable over training. Overall, the results suggest that Transformer representations are generically and persistently injective in the continuous-parameter idealization, while their practical invertibility can be probed using simple geometric diagnostics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transformer Injectivity & Geometric Robustness - Analytic Margins and Bi-Lipschitz Uniformity of Sequence-Level Hidden States von Strauss, Mikael Machine Learning Artificial Intelligence Under real-analytic assumptions on decoder-only Transformers, recent work shows that the map from discrete prompts to last-token hidden states is generically injective on finite prompt sets. We refine this picture: for each layer $\ell$ we define a collision discriminant $Δ^\ell \subset Θ$ and injective stratum $U^\ell = Θ\setminus Δ^\ell$, and prove a dichotomy -- either the model is nowhere injective on the set, or $U^\ell$ is open and dense and every $F^\ell_θ$ is injective. Under mild non-singularity assumptions on the optimizer and an absolutely continuous initialization, generic injectivity persists along smooth training trajectories over any fixed horizon. We also treat symmetry groups $G$, showing that discriminants and injective strata descend to the quotient $Θ/G$, so injectivity is naturally a property of functional equivalence classes. We complement these results with an empirical study of layerwise geometric diagnostics. We define a separation margin and a co-Lipschitz (lower Lipschitz) constant between prompt space and last-token representation space, estimated via nearest-neighbor statistics on large prompt sets. Applying these diagnostics to pretrained LLaMA-3 and Qwen models, we study behavior across layers, sequence lengths, model scales, and 8- and 4-bit activation quantization. On our sampled prompts we see no collisions in full precision or at 8 bits, while 4-bit quantization induces a small number of collisions and markedly shrinks co-Lipschitz estimates. For a small GPT-2 trained from scratch, normalized metrics remain stable over training. Overall, the results suggest that Transformer representations are generically and persistently injective in the continuous-parameter idealization, while their practical invertibility can be probed using simple geometric diagnostics. |
| title | Transformer Injectivity & Geometric Robustness - Analytic Margins and Bi-Lipschitz Uniformity of Sequence-Level Hidden States |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2511.14808 |