The Diffusion–Attention Connection
Transformers, diffusion maps, and magnetic Laplacians are different regimes of a single Markov geometry built from query–key scores — unified through the static Schrödinger bridge framework. DMAP is the equilibrium sector. Attention is NESS. The same equation governs both.
Diffusion Manifold Approximation and Projection. Coifman–Lafon diffusion maps — kernel construction, normalization, eigenvectors, and the $P^+$ operator.
Read →How diffusion maps and transformers emerge from the same Markov geometry. DMAP as equilibrium, attention as NESS, and Cartan's structure equation as the unifying object.
Read →Using diffusion coordinates as an interpretable latent space. Conditioning generative models — DiT, RFDiffusion — on physically meaningful geometry.
Read →Diffusion Transformers and flow-based generative models. How DMAP coordinates guide generation toward physically meaningful outputs.
Read →Sparse eigensolver, Nyström approximation, B300-native kernel. How DMAP achieves $O(N \cdot D)$ scaling in practice.
Read →Non-abelian SO(3) connections and vector-valued diffusion maps for protein structure space. Complement to RFDiffusion.
Row-stochastic transition matrix. Central object of DMAP — encodes manifold geometry, not sampling density.
The company equation. $F=0$ gives DMAP (equilibrium). $F \neq 0$ gives attention (NESS).
Eigenvector embedding of $P^+$ weighted by eigenvalues. Euclidean distance = diffusion distance.