Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning
math.OC, cond-mat.other, cs.LG
Submitted: 2025-07-29
Updated: 2026-09-22
Comments: 24 pages, 6 figures, 4 pseudo-code algorithms, 1 table; updated version: independent integer numbering for theorems, equations
License: http://creativecommons.org/licenses/by-nc-sa/4.0/
The gist: Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation.
Terminology
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.
Sources
- Learning with tree-based tensor formats
- Exponential Machines
- A Multi-Scale Tensor Network Architecture for Classification and Regression
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