Reverse Diffusion Sampling with Tensor Train Approximations of Hamilton–Jacobi–Bellman Equations

Abstract. Sampling from probability densities is a common challenge in fields such as uncertainty quantification and generative modeling (GM). In GM in particular, the use of reverse-time diffusion processes depending on the log densities of Ornstein–Uhlenbeck forward processes is a popular sampling tool. Berner, Richter, and Ullrich [ An Optimal Control Perspective on Diffusion-Based Generative Modeling, 2022] point out that these log densities can be obtained by the solution of a Hamilton–Jacobi–Bellman (HJB) equation known from stochastic optimal control. While this HJB equation is usually treated with indirect methods such as policy iteration and unsupervised training of black-box architectures like neural networks, we propose instead to solve the HJB equation by direct time integration, using compressed polynomials represented in the tensor train (TT) format for spatial discretization. Crucially, this method is sample-free and agnostic to normalization constants and can avoid the curse of dimensionality due to the TT compression. We provide a complete derivation of the HJB equation’s action on TT polynomials and demonstrate the performance of the proposed time-step-, rank-, and degree-adaptive integration method on a nonlinear sampling task in 20 dimensions.