Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики

Research outputpeer-review

Abstract

Low-rank matrix approximation is widely used in the analysis of big data, in recommendation systems in the Internet, for approximation solution of some equations in mechanics, and other fields. In many applications it makes sense to use matrices of unit rank for approximating since they have the simplest structure. This article provides a method for approximating positive matrices by matrices of unit rank based on the minimization of log-Chebyshev distance. The approximation problem is reduced to the optimization problem, which has a compact representation in terms of an idempotent semifield that taking maximum in the role of addition. Such semifield is often called the max-algebra. The necessary definitions and results of tropical mathematics are given and the solution of the optimization problem is derived from them. Then the solution is represented in terms of the original approximation problem. As a result, all the positive matrices which provide the minimum of approximation error are obtained in explicit form.
Original languageRussian
Title of host publicationInternational Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling
Subtitle of host publicationTechnological and Socio-Economic Processes. Proceedings
Place of PublicationSofia
PublisherScientific Technical Union of Mechanical Engineering «INDUSTRY 4.0»
Pages33-35
Volume1
Publication statusPublished - 2017
EventInternational Scientific Conference “Mathematical Modeling”
- Sofia
Duration: 13 Dec 201716 Dec 2017
http://www.mathmodel.eu

Publication series

NameInternational Scientific Conference. Mathematical Modeling.
PublisherScientific-Technical Union of Mechanical Engineering “INDUSTRY 4.0”
ISSN (Print)2535-0978
ISSN (Electronic)2603-3003

Conference

ConferenceInternational Scientific Conference “Mathematical Modeling”
Abbreviated titleMATHMODEL’ 17
CountryBulgaria
CitySofia
Period13/12/1716/12/17
Internet address

Scopus subject areas

  • Algebra and Number Theory
  • Control and Optimization

Cite this

Кривулин, Н. К., & Романова, Е. Ю. (2017). Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики. In International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling: Technological and Socio-Economic Processes. Proceedings (Vol. 1, pp. 33-35). (International Scientific Conference. Mathematical Modeling.). Sofia: Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0».
Кривулин, Николай Кимович ; Романова, Елизавета Юрьевна. / Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики. International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling: Technological and Socio-Economic Processes. Proceedings. Vol. 1 Sofia : Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0», 2017. pp. 33-35 (International Scientific Conference. Mathematical Modeling.).
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abstract = "Low-rank matrix approximation is widely used in the analysis of big data, in recommendation systems in the Internet, for approximation solution of some equations in mechanics, and other fields. In many applications it makes sense to use matrices of unit rank for approximating since they have the simplest structure. This article provides a method for approximating positive matrices by matrices of unit rank based on the minimization of log-Chebyshev distance. The approximation problem is reduced to the optimization problem, which has a compact representation in terms of an idempotent semifield that taking maximum in the role of addition. Such semifield is often called the max-algebra. The necessary definitions and results of tropical mathematics are given and the solution of the optimization problem is derived from them. Then the solution is represented in terms of the original approximation problem. As a result, all the positive matrices which provide the minimum of approximation error are obtained in explicit form.",
keywords = "idempotent mathematics, tropical mathematics, idempotent semifield, rank-one matrix approximation, log-Chebyshev distance",
author = "Кривулин, {Николай Кимович} and Романова, {Елизавета Юрьевна}",
note = "Кривулин Н. К., Романова Е. Ю. Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики // International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling. Technological and Socio-Economic Processes. Proceedings. Sofia: Scientific-Technical Union of Mechanical Engineering “INDUSTRY 4.0”, 2017. P. 33-35.",
year = "2017",
language = "русский",
volume = "1",
series = "International Scientific Conference. Mathematical Modeling.",
publisher = "Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0»",
pages = "33--35",
booktitle = "International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling",
address = "Болгария",

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Кривулин, НК & Романова, ЕЮ 2017, Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики. in International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling: Technological and Socio-Economic Processes. Proceedings. vol. 1, International Scientific Conference. Mathematical Modeling., Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0», Sofia, pp. 33-35, Sofia, 13/12/17.

Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики. / Кривулин, Николай Кимович; Романова, Елизавета Юрьевна.

International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling: Technological and Socio-Economic Processes. Proceedings. Vol. 1 Sofia : Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0», 2017. p. 33-35 (International Scientific Conference. Mathematical Modeling.).

Research outputpeer-review

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T1 - Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики

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AU - Романова, Елизавета Юрьевна

N1 - Кривулин Н. К., Романова Е. Ю. Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики // International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling. Technological and Socio-Economic Processes. Proceedings. Sofia: Scientific-Technical Union of Mechanical Engineering “INDUSTRY 4.0”, 2017. P. 33-35.

PY - 2017

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N2 - Low-rank matrix approximation is widely used in the analysis of big data, in recommendation systems in the Internet, for approximation solution of some equations in mechanics, and other fields. In many applications it makes sense to use matrices of unit rank for approximating since they have the simplest structure. This article provides a method for approximating positive matrices by matrices of unit rank based on the minimization of log-Chebyshev distance. The approximation problem is reduced to the optimization problem, which has a compact representation in terms of an idempotent semifield that taking maximum in the role of addition. Such semifield is often called the max-algebra. The necessary definitions and results of tropical mathematics are given and the solution of the optimization problem is derived from them. Then the solution is represented in terms of the original approximation problem. As a result, all the positive matrices which provide the minimum of approximation error are obtained in explicit form.

AB - Low-rank matrix approximation is widely used in the analysis of big data, in recommendation systems in the Internet, for approximation solution of some equations in mechanics, and other fields. In many applications it makes sense to use matrices of unit rank for approximating since they have the simplest structure. This article provides a method for approximating positive matrices by matrices of unit rank based on the minimization of log-Chebyshev distance. The approximation problem is reduced to the optimization problem, which has a compact representation in terms of an idempotent semifield that taking maximum in the role of addition. Such semifield is often called the max-algebra. The necessary definitions and results of tropical mathematics are given and the solution of the optimization problem is derived from them. Then the solution is represented in terms of the original approximation problem. As a result, all the positive matrices which provide the minimum of approximation error are obtained in explicit form.

KW - idempotent mathematics

KW - tropical mathematics

KW - idempotent semifield

KW - rank-one matrix approximation

KW - log-Chebyshev distance

UR - http://www.mathmodel.eu/sbornik/1-2017.pdf

M3 - статья в сборнике материалов конференции

VL - 1

T3 - International Scientific Conference. Mathematical Modeling.

SP - 33

EP - 35

BT - International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling

PB - Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0»

CY - Sofia

ER -

Кривулин НК, Романова ЕЮ. Одноранговая аппроксимация положительных матриц с использованием методов идемпотентной математики. In International Scientific Conference, 13-16 December, 2017, Borovets, Bulgaria. Mathematical Modeling: Technological and Socio-Economic Processes. Proceedings. Vol. 1. Sofia: Scientific Technical Union of Mechanical Engineering «INDUSTRY 4.0». 2017. p. 33-35. (International Scientific Conference. Mathematical Modeling.).