Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

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.

Optimization and Control (math.OC); Other Condensed Matter (cond-mat.other); Machine Learning (cs.LG)

arXiv:2507.21726 [math.OC]

(or arXiv:2507.21726v3 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2507.21726 Focus to learn more arXiv-issued DOI via DataCite

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