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CP decomposition and low-rank approximation of antisymmetric tensors

Erna Begovic, Lana Perisa

TL;DR

A suitable low-rank format is described and an alternating least squares structure-preserving algorithm for such approximation is proposed and implemented in Julia programming language and their numerical performance is discussed.

Abstract

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for finding such approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank-$1$ approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in Julia programming language and their numerical performance is discussed.

CP decomposition and low-rank approximation of antisymmetric tensors

TL;DR

A suitable low-rank format is described and an alternating least squares structure-preserving algorithm for such approximation is proposed and implemented in Julia programming language and their numerical performance is discussed.

Abstract

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for finding such approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank- approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in Julia programming language and their numerical performance is discussed.
Paper Structure (16 sections, 114 equations, 5 tables)

This paper contains 16 sections, 114 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Problem description
  4. CP-ALS with a posteriori antisymmetrization
  5. Antisymmetry-preserving CP algorithm
  6. Equivalence of the Algorithm \ref{['agm:antisymcp']} to HOPM
  7. Partial antisymmetry
  8. Ignoring the structure
  9. Preserving the structure
  10. Numerical experiments
  11. Example 1
  12. Example 2
  13. Example 3
  14. Example 4
  15. Example 5. Partial antisymmetry
  16. ...and 1 more sections

Theorems & Definitions (3)

  • proof
  • proof
  • proof