将扩散模型采样映射为量子系统的绝热演化，揭示采样极限的本质
The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport
- 构建基于学习得分的量子势能哈密顿量，实现扩散采样与量子态绝热传输的等价
- 首次建立采样精度与得分匹配误差、数据密度反庞加莱常数的定量关系
- 提供可证明的退火路径设计方法，适合理论推导和生成模型分析
我们揭示了基于得分的扩散模型采样与一类称为得分哈密顿量的薛定谔算子基态绝热传输之间的精确对应关系。该哈密顿量由学习到的得分函数构造出的量子势能构建而成。通过时变势能下的福克-普朗克方程的绝热定理，我们获得了新的密度重建边界，并提出了具有理论基础的退火调度方案。我们发现采样基本极限由得分匹配误差的平方与得分哈密顿量谱隙之比决定，即数据分布的反庞加莱常数。
原文摘要 · Abstract (English)
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.
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