证明了随机插值流的收缩性，为高维采样提供了理论支撑。
On the Contractivity of Stochastic Interpolation Flow
- 基于高斯基分布与强对数凹目标分布，推导出流映射的Lipschitz常数。
- 该常数达到Caffarelli最优传输映射的理论极限。
- 结果可推广至非高斯分布，适用于采样与估计任务。
我们研究了随机插值这一近期提出的高维采样框架，其与扩散建模有诸多相似之处。该方法通过从简单基分布中随机初始化粒子，再模拟确定性或随机动力学，在有限时间内使粒子分布收敛到目标分布。我们证明：当基分布为高斯分布、目标分布为强对数凹分布时，随机插值流映射是Lipschitz连续的，且其常数达到尖锐上界，与Caffarelli最优传输映射的理论常数一致。进一步，我们构建了非高斯分布间的Lipschitz传输映射，推广了近期关于传输方法建立函数不等式的构造。我们讨论了该定理在随机插值所需采样与估计问题中的实际意义。
原文摘要 · Abstract (English)
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.
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