Numerical Optimization for Tensor Disentanglement

TL;DR

This paper introduces Riemannian optimization techniques for tensor disentangling, aiming to reduce tensor bond dimensions efficiently, with strategies for unknown rank and hybrid methods validated through numerical experiments.

Contribution

It presents a novel Riemannian optimization framework for tensor disentangling, including a binary search for unknown ranks and hybrid approaches, advancing tensor network analysis.

Findings

Riemannian optimization effectively reduces tensor bond dimensions.

Binary search strategy helps identify optimal tensor ranks.

Hybrid methods outperform individual approaches in experiments.

Abstract

Tensor networks provide compact and scalable representations of high-dimensional data, enabling efficient computation in fields such as quantum physics, numerical partial differential equations (PDEs), and machine learning. This paper focuses on tensor disentangling, the task of identifying transformations that reduce bond dimensions by exploiting gauge freedom in the network. We formulate this task as an optimization problem over orthogonal matrices acting on a single tensor's indices, aiming to minimize the rank of its matricized form. We present Riemannian optimization methods and a joint optimization framework that alternates between optimizing the orthogonal transformation for a fixed low-rank approximation and optimizing the low-rank approximation for a fixed orthogonal transformation, offering a competitive alternative when the target rank is known. To seek the often unknown optimal rank, we introduce a binary search strategy integrated with the disentangling procedure. Numerical experiments on random tensors and tensors in an approximate isometric tensor network state are performed to compare different optimization methods and explore the possibility of combining different methods in a hybrid approach.