Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization

2026-03-12

AI Summary

This work addresses the parameter ambiguity and inefficient training inherent in conventional matrix product state (MPS)–based generative models. The authors propose a unitary MPS framework that reformulates probabilistic modeling as a manifold-constrained optimization problem and, for the first time, introduces Riemannian optimization techniques to this setting. By integrating a spatially decoupled algorithm, the method enables efficient and stable training while effectively eliminating parameter indeterminacy. Empirical evaluations on the Bars-and-Stripes and EMNIST datasets demonstrate the model’s rapid structural adaptability, superior generative performance, and favorable computational efficiency, achieving a balanced trade-off between expressivity and scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• generative modeling
• matrix product states
• Riemannian optimization
• tensor networks
• unsupervised learning

Innovation

• unitary matrix product states
• Riemannian optimization
• tensor networks
• generative modeling
• manifold constraints