Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

TL;DR

This paper introduces a geometric framework and new optimization algorithms for Tree Tensor Networks, enhancing their training efficiency and applicability in machine learning tasks.

Contribution

It provides a formal differential geometry analysis of TTNs and develops novel first- and second-order optimization algorithms tailored to their structure.

Findings

Efficient optimization algorithms for TTNs are developed.

Numerical experiments demonstrate improved training performance.

The methods enable effective kernel learning with TTNs.

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 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.