Sampling in High-Dimensions using Stochastic Interpolants and Forward-Backward Stochastic Differential Equations
This work addresses the challenge of efficient sampling in high-dimensional spaces for applications in machine learning and statistics, representing a novel method for a known bottleneck.
The authors tackled the problem of sampling from high-dimensional probability distributions by developing diffusion-based algorithms that transport samples from a Gaussian to a target distribution in finite time, using stochastic interpolants and solving Hamilton-Jacobi-Bellman PDEs with FBSDE and machine learning methods, and demonstrated effectiveness in numerical experiments where conventional methods struggle.
We present a class of diffusion-based algorithms to draw samples from high-dimensional probability distributions given their unnormalized densities. Ideally, our methods can transport samples from a Gaussian distribution to a specified target distribution in finite time. Our approach relies on the stochastic interpolants framework to define a time-indexed collection of probability densities that bridge a Gaussian distribution to the target distribution. Subsequently, we derive a diffusion process that obeys the aforementioned probability density at each time instant. Obtaining such a diffusion process involves solving certain Hamilton-Jacobi-Bellman PDEs. We solve these PDEs using the theory of forward-backward stochastic differential equations (FBSDE) together with machine learning-based methods. Through numerical experiments, we demonstrate that our algorithm can effectively draw samples from distributions that conventional methods struggle to handle.