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A Numerical Solution to KPD

Daizhan Cheng

Abstract

A stationary value based algorithm (SVA) is provided to solve the nearest Kronecker product decomposition (KPD) problem of vector form hypermatrices. Using the algorithm successively, the finite sum KPD is also solved. Then the permutation matrix is introduced. Using it, the KPD of matrix form hypermatrices is converted to its equivalent KPD of vector forms, and then the SVA is also applicable to solve the same problems for vector form hypermatrix. Some numerical examples are presented to demonstrate the new algorithm and to compare it with existing methods.

A Numerical Solution to KPD

Abstract

A stationary value based algorithm (SVA) is provided to solve the nearest Kronecker product decomposition (KPD) problem of vector form hypermatrices. Using the algorithm successively, the finite sum KPD is also solved. Then the permutation matrix is introduced. Using it, the KPD of matrix form hypermatrices is converted to its equivalent KPD of vector forms, and then the SVA is also applicable to solve the same problems for vector form hypermatrix. Some numerical examples are presented to demonstrate the new algorithm and to compare it with existing methods.
Paper Structure (13 sections, 16 theorems, 84 equations)

This paper contains 13 sections, 16 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-Tensor Product of Matrices
  4. Hypermatrix
  5. Hypervectors
  6. KPD of Vector Form Hypermatrices
  7. Exact KPD of Vector Form Hypermatrices
  8. NKP of Vector Form Hypermatrices
  9. Finite Sum KPD via NKP of Vector Form Hypermatrices
  10. KPD for Matrix Form Hypermaxices
  11. Matrix Form Decomposability
  12. A Numerical Example
  13. Concludin Remarks

Key Result

Proposition 2.2

Theorems & Definitions (34)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Remark 2.10
  • ...and 24 more