用少量样本零梯度推导概率分布间的最优传输映射
In-Context Operator Learning on the Space of Probability Measures
- 在不更新参数前提下，仅凭少量样本即可学习分布间传输映射
- 当任务集中在低维流形时，泛化误差随提示规模和模型容量改善
- 对高斯等特定分布可精确恢复传输映射，适合生成建模场景
我们提出在概率测度空间上的上下文操作符学习方法，用于最优传输（OT）。目标是学习一个单一解算符，仅通过每分布的少量样本作为提示，在推理时不进行梯度更新，直接将一对分布映射为最优传输映射。我们对解算符进行参数化，并在两种情形下建立缩放定律：在非参数设置中，当任务集中于源-目标对的低内在维流形上时，建立了泛化误差界，量化了上下文精度如何随提示大小、任务内在维度和模型容量变化；在参数设置（如高斯族）中，给出了能精确恢复最优传输映射的显式架构，并提供有限样本下的过风险界。在合成传输与生成建模基准上的数值实验验证了该框架的有效性。
原文摘要 · Abstract (English)
We introduce \emph{in-context operator learning on probability measure spaces} for optimal transport (OT). The goal is to learn a single solution operator that maps a pair of distributions to the OT map, using only few-shot samples from each distribution as a prompt and \emph{without} gradient updates at inference. We parameterize the solution operator and develop scaling-law theory in two regimes. In the \emph{nonparametric} setting, when tasks concentrate on a low-intrinsic-dimension manifold of source--target pairs, we establish generalization bounds that quantify how in-context accuracy scales with prompt size, intrinsic task dimension, and model capacity. In the \emph{parametric} setting (e.g., Gaussian families), we give an explicit architecture that recovers the exact OT map in context and provide finite-sample excess-risk bounds. Our numerical experiments on synthetic transports and generative-modeling benchmarks validate the framework.
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