MERACLE: Constructive Layer-Wise Conversion of a Tensor Train into a MERA

Kim Batselier*, Andrzej Cichocki, Ngai Wong

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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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.

Original languageEnglish
Pages (from-to)257-279
JournalCommunications on Applied Mathematics and Computation
Volume3
Issue number2
DOIs
Publication statusPublished - 2021

Keywords

  • Disentangler
  • HOSVD
  • MERA
  • Tensor train
  • Tensors
  • Tucker decomposition

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