Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization
This work addresses the training inefficiency of tensor networks for unsupervised learning, which is an incremental improvement for the machine learning community.
The paper tackles the problem of inefficient training of matrix product states (MPS) for generative modeling by developing a Riemannian optimization approach with a space-decoupling algorithm, resulting in fast adaptation, stable updates, and strong performance on Bars-and-Stripes and EMNIST datasets.
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.