Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

Marius Willner; Marco Trenti; Dirk Lebiedz

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arXiv · 2507.21726

Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

Marius Willner·Marco Trenti·Dirk Lebiedz

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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 quotient geometry underlying the TTN parameter space. Our framework allows for arbitrary horizontal distributions, and we develop efficient first- and second-order optimization algorithms that exploit this geometry. Additionally, we devise a backpropagation algorithm for training TTNs in a kernel learning setting. We validate our methods through numerical experiments on a representative digit classification task and reveal an important tradeoff between two different horizontal distributions that are available for TTNs: while one offers cleaner geometric statements, the other ultimately leads to more efficient algorithms.

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Marius Willner, Marco Trenti, Dirk Lebiedz. 2026-09-22. Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning. https://arxiv.org/abs/2507.21726

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