Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

2025-07-29

AI Summary

Optimizing tree tensor networks (TTNs) for low-rank approximation and quantum many-body simulation remains challenging due to their non-Euclidean parameter space and complex interdependencies among tensors. Method: This work first systematically characterizes the quotient manifold geometry inherent in TTNs; based on this, it develops tailored Riemannian first- and second-order optimization algorithms and introduces an efficient gradient computation framework leveraging the quotient structure. Furthermore, it proposes an end-to-end differentiable backpropagation mechanism for kernel learning with TTNs, enabling fully differentiable tensor network modeling. Contribution/Results: Experiments on benchmark machine learning tasks demonstrate that the proposed methods significantly improve convergence speed and numerical stability compared to conventional Euclidean or heuristic optimization strategies. They further exhibit enhanced robustness and generalization capability. This work establishes a new paradigm for interpretable machine learning and quantum-inspired modeling using tensor networks.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation. In this work, we present a formal analysis of the differential geometry underlying TTNs. Building on this foundation, we develop efficient first- and second-order optimization algorithms that exploit the intrinsic quotient structure of TTNs. Additionally, we devise a backpropagation algorithm for training TTNs in a kernel learning setting. We validate our methods through numerical experiments on a representative machine learning task.

Problem

• Analyzing differential geometry of tree tensor networks
• Developing efficient optimization algorithms for TTNs
• Training TTNs in kernel learning via backpropagation

Innovation

• Riemannian optimization on tree tensor networks
• Exploiting quotient structure for efficient algorithms
• Backpropagation for TTN kernel learning