TY - JOUR

T1 - On the rank-one approximation of positive matrices using tropical optimization methods

AU - Krivulin, N. K.

AU - Romanova, E. Yu.

N1 - Krivulin, N.K. & Romanova, E.Y. Vestnik St.Petersb. Univ.Math. (2019) 52: 145. https://doi.org/10.1134/S1063454119020080

PY - 2019

Y1 - 2019

N2 - An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.

AB - An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.

KW - tropical mathematics

KW - tropical optimization

KW - max-algebra

KW - rank-one matrix approximation

KW - log-Chebyshev distance function

UR - http://www.scopus.com/inward/record.url?scp=85067190825&partnerID=8YFLogxK

U2 - 10.1134/S1063454119020080

DO - 10.1134/S1063454119020080

M3 - Article

VL - 52

SP - 145

EP - 153

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -