基于黎曼几何优化树张量网络，提升机器学习效率
Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning
- 利用树状张量网络的商流形结构设计优化算法
- 支持一阶与二阶优化，适用于核学习场景
- 适合需高效低秩表示的机器学习任务
树张量网络（TTNs）广泛应用于低秩逼近和量子多体模拟。本文首次对TTNs的微分几何结构进行形式化分析，并在此基础上构建了高效的前向与二阶优化算法，充分利用其内在商流形结构。此外，我们提出一种用于核学习设置中训练TTNs的反向传播算法。通过在典型机器学习任务上的数值实验验证了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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