We investigate stochastic interpolation, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling. Stochastic interpolation generates a data sample by first randomly initializing a particle drawn from a simple base distribution, then simulating deterministic or stochastic dynamics such that in finite time the particle's distribution converges to the target. We show that for a Gaussian base distribution and a strongly log-concave target distribution, the stochastic interpolation flow map is Lipschitz with a sharp constant which matches that of Caffarelli's theorem for optimal transport maps. We are further able to construct Lipschitz transport maps between non-Gaussian distributions, generalizing some recent constructions in the literature on transport methods for establishing functional inequalities. We discuss the practical implications of our theorem for the sampling and estimation problems required by stochastic interpolation.

翻译：本文研究随机插值——一种近期提出的高维采样框架，其与扩散模型具有诸多相似性。随机插值通过以下步骤生成数据样本：首先从简单基分布中随机初始化粒子，随后模拟确定性或随机动力学，使得粒子分布在有限时间内收敛至目标分布。我们证明，对于高斯基分布与强对数凹目标分布，随机插值流映射具有利普希茨连续性，其锐利常数与卡法雷利定理中关于最优传输映射的常数一致。我们进一步能够构建非高斯分布间的利普希茨传输映射，从而推广了近期文献中关于建立函数不等式的传输方法中的若干构造。本文还讨论了该定理对随机插值所需采样与估计问题的实际意义。