We propose a hierarchical tensor-network approach for approximating high-dimensional probability density via empirical distribution. This leverages randomized singular value decomposition (SVD) techniques and involves solving linear equations for tensor cores in this tensor network. The complexity of the resulting algorithm scales linearly in the dimension of the high-dimensional density. An analysis of estimation error demonstrates the effectiveness of this method through several numerical experiments.

翻译：本文提出一种分层张量网络方法，通过经验分布近似高维概率密度。该方法利用随机化奇异值分解技术，并涉及求解该张量网络中张量核的线性方程组。所得算法的复杂度随高维密度维度线性增长。通过多项数值实验对估计误差进行分析，验证了该方法的有效性。