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：我们提出了一种层次化张量网络方法，用于通过经验分布近似高维概率密度函数。该方法利用随机奇异值分解技术，并通过求解张量网络中张量核所对应的线性方程实现。所提出算法的复杂度与高维密度的维数呈线性关系。通过若干数值实验的估计误差分析表明，该方法具备良好的有效性。