提出量子势估计的样本复杂度分析，证明高精度估计在无限样本下仍可行。
Sample complexity of Schrödinger potential estimation
- 基于经验KL风险最小化估计量子势，通过理论推导实现非渐近上界。
- 样本量n增大时，超出风险以O(log²n/n)速度下降，即使分布支持集无界。
- 适用于生成模型中的扩散路径设计，尤其适合研究理论性能边界者。
我们研究了基于薛定谔桥与随机最优控制的生成建模中关键的薛定谔势估计问题。给定一个简单的先验扩散过程，该方法需寻找连接两个给定分布ρ₀和ρₜ*的最小代价路径，其最优漂移可由薛定谔势表示。本文在合理假设下，分析了对一类可接受的对数势进行经验KL风险最小化的泛化能力，目标是拟合时间T的边缘分布。我们推导出ρₜ*与估计对数势对应的终端密度之间KL散度的非渐近高概率上界。特别地，当样本量n趋于无穷时，即使ρ₀和ρₜ*均具有无界支撑，超出的KL风险仍可达到O(log²n/n)的收敛速率。
原文摘要 · Abstract (English)
We address the problem of Schrödinger potential estimation, which plays a crucial role in modern generative modelling approaches based on Schrödinger bridges and stochastic optimal control for SDEs. Given a simple prior diffusion process, these methods search for a path between two given distributions $ρ_0$ and $ρ_T^*$ requiring minimal efforts. The optimal drift in this case can be expressed through a Schrödinger potential. In the present paper, we study generalization ability of an empirical Kullback-Leibler (KL) risk minimizer over a class of admissible log-potentials aimed at fitting the marginal distribution at time $T$. Under reasonable assumptions on the target distribution $ρ_T^*$ and the prior process, we derive a non-asymptotic high-probability upper bound on the KL-divergence between $ρ_T^*$ and the terminal density corresponding to the estimated log-potential. In particular, we show that the excess KL-risk may decrease as fast as $O(\log^2 n / n)$ when the sample size $n$ tends to infinity even if both $ρ_0$ and $ρ_T^*$ have unbounded supports.
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