Title: On the Contractivity of Stochastic Interpolation Flow Show Chinese title

Title: 关于随机插值流的收缩性

Abstract: 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. Show Chinese abstract

Abstract: 我们研究了随机插值，这是一种最近引入的高维采样框架，与扩散模型有许多相似之处。 随机插值通过首先从一个简单的基础分布中随机初始化一个粒子，然后模拟确定性或随机动力学，使得在有限时间内粒子的分布收敛到目标分布，从而生成一个数据样本。 我们证明了对于高斯基础分布和强对数凹目标分布，随机插值流映射是具有尖锐常数的利普希茨映射，该常数与Caffarelli关于最优传输映射定理中的常数相匹配。 我们还能够构造非高斯分布之间的利普希茨传输映射，推广了一些关于建立函数不等式的传输方法文献中的最新构造。 我们讨论了我们的定理对随机插值所需的采样和估计问题的实际意义。