Sampling from probability densities is a common challenge in fields such as Uncertainty Quantification (UQ) and Generative Modelling (GM). In GM in particular, the use of reverse-time diffusion processes depending on the log-densities of Ornstein-Uhlenbeck forward processes are a popular sampling tool. In Berner et al. [2022] the authors point out that these log-densities can be obtained by solution of a \textit{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, 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 Tensor Train polynomials and demonstrate the performance of the proposed time-step-, rank- and degree-adaptive integration method on a nonlinear sampling task in 20 dimensions.

翻译：从概率密度中采样是不确定性量化（UQ）和生成式建模（GM）等领域的常见挑战。特别是在生成式建模中，基于奥恩斯坦-乌伦贝克前向过程对数密度的反向时间扩散过程是一种流行的采样工具。在 Berner 等人 [2022] 的研究中，作者指出这些对数密度可通过求解随机最优控制中著名的哈密顿-雅可比-贝尔曼（HJB）方程获得。虽然该 HJB 方程通常采用间接方法处理，如策略迭代和神经网络等黑箱架构的无监督训练，我们转而提出通过直接时间积分求解 HJB 方程，并使用以张量列（TT）格式表示的压缩多项式进行空间离散化。关键的是，该方法无样本需求、与归一化常数无关，且由于张量列压缩能够避免维度灾难。我们提供了 HJB 方程对张量列多项式作用的完整推导，并展示了所提出的时间步长、秩和阶自适应积分方法在 20 维非线性采样任务上的性能。