Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization

TL;DR

This paper introduces a Riemannian optimization method for unitary matrix product states, enhancing generative modeling efficiency and stability in high-dimensional probability distributions, with promising results on benchmark datasets.

Contribution

It develops a Riemannian optimization approach for unitary MPS, improving training efficiency and stability in generative modeling tasks.

Findings

Fast adaptation to data structure

Stable parameter updates

Strong performance on benchmark datasets

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.