The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport
arXiv:2606.05217
The paper establishes an exact link between score‑based diffusion model sampling and adiabatic transport of ground states of specially constructed Schrödinger operators, providing new density reconstruction bounds and annealing schedules based on spectral properties.
Abstract
We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincaré constant of the data density.
26 pages, 10 figures. v3: Expanded supplementary discussion in appendix of application to various score-based parameterizations. Main results and main text unchanged