Efficient probability density estimation is a core challenge in statistical machine learning. Tensor-based probabilistic graph methods address interpretability and stability concerns encountered in neural network approaches. However, a substantial number of potential tensor permutations can lead to a tensor network with the same structure but varying expressive capabilities. In this paper, we take tensor ring decomposition for density estimator, which significantly reduces the number of permutation candidates while enhancing expressive capability compared with existing used decompositions. Additionally, a mixture model that incorporates multiple permutation candidates with adaptive weights is further designed, resulting in increased expressive flexibility and comprehensiveness. Different from the prevailing directions of tensor network structure/permutation search, our approach provides a new viewpoint inspired by ensemble learning. This approach acknowledges that suboptimal permutations can offer distinctive information besides that of optimal permutations. Experiments show the superiority of the proposed approach in estimating probability density for moderately dimensional datasets and sampling to capture intricate details.

翻译：高效的概率密度估计是统计机器学习中的核心挑战。基于张量的概率图方法解决了神经网络方法中遇到的可解释性和稳定性问题。然而，大量潜在的张量排列可能导致具有相同结构但表达能力各异的张量网络。在本文中，我们采用张量环分解进行密度估计，与现有分解方法相比，该方法显著减少了排列候选数量，同时增强了表达能力。此外，我们进一步设计了一个包含多个排列候选及其自适应权重的混合模型，从而提高了表达的灵活性和全面性。与当前主流的张量网络结构/排列搜索方向不同，我们的方法提供了受集成学习启发的新视角。该方法承认次优排列除了包含最优排列的信息外，还能提供独特信息。实验表明，所提方法在中等维度数据集的概率密度估计以及捕捉复杂细节的采样方面具有优越性。