# Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization

arXiv: https://arxiv.org/abs/2603.12026 · HTML: https://mlsanity.com/paper/2603.12026/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 41 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 20 of 20)

- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines](https://mlsanity.com/paper/1712.04144/) (2017)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Step Size Matters in Deep Learning](https://mlsanity.com/paper/1805.08890/) (2018)
- [Parallelized Tensor Train Learning of Polynomial Classifiers](https://mlsanity.com/paper/1612.06505/) (2016)
- [Generative modeling with projected entangled-pair states](https://mlsanity.com/paper/2202.08177/) (2022)
- [DeepTensor: Low-Rank Tensor Decomposition with Deep Network Priors](https://mlsanity.com/paper/2204.03145/) (2022)
- [Probabilistic Modeling with Matrix Product States](https://mlsanity.com/paper/1902.06888/) (2019)
- [Structured Low-Rank Algorithms: Theory, MR Applications, and Links to Machine Learning](https://mlsanity.com/paper/1910.12162/) (2019)
- [A general framework of Riemannian adaptive optimization methods with a convergence analysis](https://mlsanity.com/paper/2409.00859/) (2024)
- [Deep Tree Tensor Networks](https://mlsanity.com/paper/2502.09928/) (2025)
- [Shortcut Matrix Product States and its applications](https://mlsanity.com/paper/1812.05248/) (2018)
- [Tensor Star Tensor Decomposition and Its Applications to Higher-order Compression and Completion](https://mlsanity.com/paper/2403.10481/) (2024)
- [A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints](https://mlsanity.com/paper/2501.13830/) (2025)
- [Normalized tensor train decomposition](https://mlsanity.com/paper/2511.04369/) (2025)

## Abstract (arXiv)

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.