The Diffusion-Attention Connection

Abstract:Softmax attention is the row-normalized operator of a diffusion map: both normalize a learned score into a Markov operator, and differ only in what the score is allowed to contain. Decomposing that score reveals three geometric sectors: a metric core with Witten-Laplacian continuum limit, an exact node-potential sector corresponding to a Markov--Witten change of measure, and a circulating sector realized as irreversible Markov--Girsanov transport or as a magnetic $U(1)$ phase. Attention thereby becomes auditable in familiar mathematics: every trained head carries measurable geometry, potential, and flux, while standard mechanisms acquire geometric addresses---Coifman--Lafon normalization as an exact density correction, rotary embeddings as pure gauge, and AdaLN as a Cauchy--Green deformation combined with an Witten deformation. Experiments on pretrained diffusion transformers and language models test this decomposition: enforcing positive-semi-definite geometry is nearly free, consistent with the identification, whereas removing circulation incurs a substantial cost, sharpest on induction.

Machine Learning (cs.LG)

arXiv:2604.09560 [cs.LG]

(or arXiv:2604.09560v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2604.09560 Focus to learn more arXiv-issued DOI via DataCite

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