aiqbench analysis
This research establishes a rigorous mathematical link between score-based diffusion sampling and adiabatic quantum transport, deriving fundamental limits for sampling efficiency based on the spectral gap of the underlying Score Hamiltonian. By framing the process through Schrödinger operators and Fokker-Planck equations, the work provides principled annealing schedules and density reconstruction bounds that could optimize the theoretical performance of generative models. The study bridges advanced mathematical physics with machine learning, offering a new analytical framework for understanding the convergence properties of diffusion models.
The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport
arXiv:2606.05217v1 Announce Type: cross Abstract: We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schr\"odinger 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 Poincar\'e constant of the data density.
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.
Mathematical Physics (math-ph); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Data Analysis, Statistics and Probability (physics.data-an)
arXiv:2606.05217 [math-ph]
(or arXiv:2606.05217v1 [math-ph] for this version)
Full text under the publisher's open licence · 106 words
