# Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

arXiv: https://arxiv.org/abs/2507.21726 · HTML: https://mlsanity.com/paper/2507.21726/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 48 references that are in the map.
- First posted on arXiv 29 Jul 2025.

## Builds on (most cited first, 8 of 8)

- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [An Overview of Neural Network Compression](https://mlsanity.com/paper/2006.03669/) (2020)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [Quantum-inspired Machine Learning on high-energy physics data](https://mlsanity.com/paper/2004.13747/) (2020)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [Tensor-based algorithms for image classification](https://mlsanity.com/paper/1910.02150/) (2019)
- [Learning with tree-based tensor formats](https://mlsanity.com/paper/1811.04455/) (2018)

## Built on by (most cited first, 3 of 3)

- [Natural Riemannian gradient for learning functional tensor networks](https://mlsanity.com/paper/2604.09263/) (2026)
- [Quantum-Inspired Robust and Scalable SAR Object Classification](https://mlsanity.com/paper/2604.25755/) (2026)
- [Stochastic Optimization of Tree Tensor Networks](https://mlsanity.com/paper/2609.00870/) (2026)

## Abstract (arXiv)

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.