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This research establishes a rigorous mathematical bridge between score-based diffusion models and quantum adiabatic transport by defining "Score Hamiltonians." By framing generative sampling as the evolution of ground states in Schrödinger operators with time-varying potentials, the study derives fundamental limits on sampling efficiency based on spectral gaps rather than heuristic annealing schedules. This work provides a principled theoretical foundation for optimizing diffusion processes using tools from mathematical physics and quantum mechanics.

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The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport

arXiv31 iulie 20261 min de cititArticolul și perspectiva sunt în engleză.

arXiv:2606.05217v4 Announce Type: replace-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.

26 pages, 10 figures. v3: Expanded supplementary discussion in appendix of application to various score-based parameterizations. Main results and main text unchanged

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:2606.05217v4 [math-ph] for this version)

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