Rank-One Approximation of Positive Matrices Based on Methods of Tropical Mathematics
| Autor: | E. Yu. Romanova, Nikolai Krivulin |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Optimization problem
Basis (linear algebra) Rank (linear algebra) General Mathematics 010102 general mathematics General Physics and Astronomy 010103 numerical & computational mathematics 01 natural sciences Matrix (mathematics) Approximation error Idempotence Applied mathematics 0101 mathematics Representation (mathematics) Semifield Mathematics |
| Zdroj: | Vestnik St. Petersburg University, Mathematics. 51:133-143 |
| ISSN: | 1934-7855 1063-4541 |
| DOI: | 10.3103/s106345411802005x |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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