Transformer Injectivity & Geometric Robustness - Analytic Margins and Bi-Lipschitz Uniformity of Sequence-Level Hidden States

TL;DR

This paper investigates the injectivity and geometric robustness of Transformer models, providing theoretical conditions for injectivity and empirical diagnostics to assess invertibility and stability of sequence representations.

Contribution

It introduces a formal framework for analyzing injectivity in Transformers, including a dichotomy theorem and geometric diagnostics, with empirical validation on pretrained models.

Findings

Transformers are generically injective in the continuous-parameter limit.

Quantization at 4 bits induces small collisions and reduces geometric robustness.

Layerwise diagnostics reveal stability of representations across training and model scales.

Abstract

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 $\Delta^\ell \subset \Theta$ and injective stratum $U^\ell = \Theta \setminus \Delta^\ell$, and prove a dichotomy -- either the model is nowhere injective on the set, or $U^\ell$ is open and dense and every $F^\ell_\theta$ 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 $\Theta/G$, 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.