MERACLE: Constructive layer-wise conversion of a Tensor Train into a MERA

TL;DR

This paper introduces two efficient algorithms for converting tensor trains into Tucker and MERA formats, automatically determining ranks and improving storage efficiency, with a novel iterative method for optimal disentangler tensors.

Contribution

The paper presents new algorithms for tensor train to MERA conversion that are computationally efficient, automatically rank-determining, and include a novel iterative method for optimal disentangler tensors.

Findings

Algorithms effectively convert tensor trains to Tucker and MERA formats.

The methods automatically determine multilinear and MERA ranks.

Numerical experiments show storage benefits of low-rank MERA.

Abstract

In this article two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz (MERA). The Tucker core tensor is never explicitly computed but stored as a tensor train instead, resulting in both computationally and storage efficient algorithms. Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error. In addition, an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors, which are a crucial component in the construction of a low-rank MERA. Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.