Tensor networks, which are originally developed for characterizing complex quantum many-body systems, have recently emerged as a powerful framework for capturing high-dimensional probability distributions with strong physical interpretability. This paper systematically studies matrix product states (MPS) for generative modeling and shows that unitary MPS, which is a tensor-network architecture that is both simple and expressive, offers clear benefits for unsupervised learning by reducing ambiguity in parameter updates and improving efficiency. To overcome the inefficiency of standard gradient-based MPS training, we develop a Riemannian optimization approach that casts probabilistic modeling as an optimization problem with manifold constraints, and further derive an efficient space-decoupling algorithm. Experiments on Bars-and-Stripes and EMNIST datasets demonstrate fast adaptation to data structure, stable updates, and strong performance while maintaining the efficiency and expressive power of MPS.

翻译：张量网络最初为描述复杂量子多体系统而发展，近年来已成为捕捉高维概率分布且具有强物理可解释性的强大框架。本文系统研究了矩阵乘积态（MPS）在生成建模中的应用，证明酉矩阵乘积态作为一种简洁且表达能力强的张量网络架构，通过减少参数更新的模糊性并提升效率，为无监督学习带来显著优势。为克服基于标准梯度的MPS训练效率低下的问题，我们开发了一种黎曼优化方法，将概率建模转化为具有流形约束的优化问题，并进一步推导出高效的空间解耦算法。在Bars-and-Stripes和EMNIST数据集上的实验表明，该方法能快速适应数据结构、实现稳定更新，在保持MPS效率与表达能力的同時展现出优越性能。