用黎曼优化提升酉矩阵乘积态生成模型效率与稳定性
Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization
- 采用酉矩阵乘积态结构，减少参数更新歧义
- 在巴-斯特拉数据集上实现快速结构适应与稳定训练
- 适合追求高效高表达力生成建模的研究者
张量网络最初用于描述复杂的量子多体系统，近年来成为捕捉高维概率分布的强大框架，兼具物理可解释性。本文系统研究了矩阵乘积态（MPS）在生成建模中的应用，发现酉矩阵乘积态（unitary MPS）这一结构简洁且表达力强，能显著提升无监督学习的效率并减少参数更新歧义。为克服传统基于梯度的MPS训练效率低下问题，提出一种黎曼优化方法，将概率建模转化为带流形约束的优化问题，并推导出高效的解耦算法。在Bars-and-Stripes和EMNIST数据集上的实验表明，该方法具备快速适应数据结构、参数更新稳定、性能优异的特点，同时保持MPS原有的高效性与表达力。
原文摘要 · Abstract (English)
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.
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