The Depth Flow of Token Representations Is Nonlinear and Does Not Descend Its Own Density
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Computer Science > Computation and Language
arXiv:2608.29706 (cs)
[Submitted on 30 Aug 2026]
Title:The Depth Flow of Token Representations Is Nonlinear and Does Not Descend Its Own Density
Authors:Alexandre Quemy
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Abstract:A token's representation is carried through the network layer by layer. The whole vocabulary carried together forms a flow. We fit this flow's equation of motion as a discrete Langevin model over corpus-mean trajectories of Pythia-160M and Pythia-410M, and score the predicted steps on held-out tokens. Linear maps are often used as cheap surrogates for a layer. The flow they summarize is not linear: a quadratic drift beats the linear linear map at every transition of both models, and the Kramers--Moyal estimator agrees wherever its neighborhoods stay local.
We then characterize the flow further. First, we show that it does not descend its own log-density. The drift instead descends a potential that is not the density. Second, the rotational component is not negligible, $4$ to $45\%$ of the explainable drift, and the circulation shows in what the flow preserves: a token keeps its angular rank across all thirteen layers while its norm rank is shuffled and its concentration rank is reversed by the last block.
| Subjects: | Computation and Language (cs.CL) |
| Cite as: | arXiv:2608.29706 [cs.CL] |
| (or arXiv:2608.29706v1 [cs.CL] for this version) | |
| https://doi.org/10.48550/arXiv.2608.29706
arXiv-issued DOI via DataCite (pending registration)
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View a PDF of the paper titled The Depth Flow of Token Representations Is Nonlinear and Does Not Descend Its Own Density, by Alexandre Quemy
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