mlsanityData to 14 Oct 2026

The Depth Flow of Token Representations Is Nonlinear and Does Not Descend Its Own Density

arXiv 2608.29706 ↗First posted 30 Aug 2026cs.CLcited by 0

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Abstract (arXiv)

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.

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References · 20

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
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  12. 2021Sinkformers: Transformers with Doubly Stochastic Attentioncited by 138
8 more
  1. 2023Jump to Conclusions: Short-Cutting Transformers With Linear Transformationscited by 90
  2. 2017Residual Connections Encourage Iterative Inferencecited by 77
  3. 2022Outliers Dimensions that Disrupt Transformers Are Driven by Frequencycited by 55
  4. 2025Transformer Dynamics: A neuroscientific approach to interpretability of large language modelscited by 11
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  6. 2024Lines of Thought in Large Language Modelscited by 5
  7. 2026Dynamics of the Transformer Residual Stream: Coupling Spectral Geometry to Network Topologycited by 5
  8. 2026A Geometric Perspective on Next-Token Prediction in Large Language Models: Three Emerging Phasescited by 3

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