### Abstract

Original language | English |
---|---|

Pages (from-to) | 133-143 |

Number of pages | 11 |

Journal | Vestnik St. Petersburg University: Mathematics |

Volume | 51 |

Issue number | 2 |

Early online date | 16 Jun 2018 |

DOIs | |

Publication status | Published - 2018 |

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### Scopus subject areas

- Control and Optimization
- Management Science and Operations Research
- Algebra and Number Theory

### Cite this

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**Rank-one approximation of positive matrices based on methods of tropical mathematics.** / Кривулин, Николай Кимович; Романова, Елизавета Юрьевна.

Research output

TY - JOUR

T1 - Rank-one approximation of positive matrices based on methods of tropical mathematics

AU - Кривулин, Николай Кимович

AU - Романова, Елизавета Юрьевна

PY - 2018

Y1 - 2018

N2 - Low-rank matrix approximation finds wide application in the analysis of big data, in recommendation systems on the Internet, for the approximate solution of some equations of mechanics, and in other fields. In this paper, a method for approximating positive matrices by rank-one matrices on the basis of minimization of log-Chebyshev distance is proposed. The problem of approximation reduces to an optimization problem having a compact representation in terms of an idempotent semifield in which the operation of taking the maximum plays the role of addition and which is often referred to as max-algebra. The necessary definitions and preliminary results of tropical mathematics are given, on the basis of which the solution of the original problem is constructed. Using the methods and results of tropical optimization, all positive matrices at which the minimum of approximation error is reached are found in explicit form. A numerical example illustrating the application of the rank-one approximation is considered.

AB - Low-rank matrix approximation finds wide application in the analysis of big data, in recommendation systems on the Internet, for the approximate solution of some equations of mechanics, and in other fields. In this paper, a method for approximating positive matrices by rank-one matrices on the basis of minimization of log-Chebyshev distance is proposed. The problem of approximation reduces to an optimization problem having a compact representation in terms of an idempotent semifield in which the operation of taking the maximum plays the role of addition and which is often referred to as max-algebra. The necessary definitions and preliminary results of tropical mathematics are given, on the basis of which the solution of the original problem is constructed. Using the methods and results of tropical optimization, all positive matrices at which the minimum of approximation error is reached are found in explicit form. A numerical example illustrating the application of the rank-one approximation is considered.

KW - tropical mathematics

KW - idempotent semifield

KW - rank-one matrix approximation

KW - log-Chebyshev distance function

U2 - 10.3103/S106345411802005X

DO - 10.3103/S106345411802005X

M3 - Article

VL - 51

SP - 133

EP - 143

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -