Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning
This work addresses optimization challenges for TTNs, which are used in low-rank approximation and quantum simulation, but it appears incremental as it builds on existing geometric foundations.
The authors tackled the problem of optimizing tree tensor networks (TTNs) by developing efficient first- and second-order algorithms based on a formal analysis of their differential geometry, and validated these methods with numerical experiments on a machine learning task.
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.