# The Diffusion-Attention Connection

arXiv: https://arxiv.org/abs/2604.09560 · HTML: https://mlsanity.com/paper/2604.09560/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2026.

## Built on by (most cited first, 1 of 1)

- [FlashDiffusion: Fused Tiled Kernel Spectral Decomposition](https://mlsanity.com/paper/2609.38198/) (2026)

## Abstract (arXiv)

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.