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.
@article{arxiv.2507.21726,
title = {Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning},
author = {Marius Willner and Marco Trenti and Dirk Lebiedz},
journal= {arXiv preprint arXiv:2507.21726},
year = {2025}
}
Comments
24 pages, 6 figures, 4 pseudo-code algorithms, 1 table; updated version: additional explanation for computational advantages of Cart. horiz. space in Sec. 6; updated Fig. 6 accordingly; fixed typos and added references