Algebraic Solution of Tropical Best Approximation Problems

Krivulin, Nikolai

doi:10.3390/math11183949

Cited by 4 publications

(3 citation statements)

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“…To represent the system of equations at (9) in vector form, we follow the approach in Krivulin (2023) to introduce the following vector-matrix notation:…”

Section: Vector Form Of Approximation Problemmentioning

confidence: 99%

“…The discrete best approximation problems of functions, which appear in recent works on neural networks and machine learning (see, e.g., Zhang et al, 2018;Maragos et al, 2021), constitute another direction of tropical approximation that needs further research. As an attempt to address this need, a general problem of discrete best approximation of a function is introduced and examined in tropical algebra settings in Krivulin (2023), where a function defined on a tropical semifield is approximated by tropical Puiseux polynomials and rational functions with respect to a generalized metric.…”

Section: Introductionmentioning

confidence: 99%

“…We further develop the approach described in Krivulin (2023), where the exponents (powers) are assumed to be fixed in advance, and the solution concentrates on evaluating the coefficients of the monomials in the approximating polynomial. Since the situation when the exponents are also unknown is much more common, we now solve a more general problem where the construction of an approximating polynomial involves the evaluation of both unknown coefficient and exponent of each monomial.…”

Section: Introductionmentioning

confidence: 99%

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On Solution of Tropical Discrete Best Approximation Problems

Krivulin

2024

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We consider a discrete best approximation problem formulated in the framework of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. Given a set of samples of input and output of an unknown function, the problem is to construct a generalized tropical Puiseux polynomial that best approximates the function in the sense of a tropical distance function. The construction of an approximate polynomial involves the evaluation of both unknown coefficient and exponent of each monomial in the polynomial. To solve the approximation problem, we first reduce the problem to an equation in unknown vector of coefficients, which is given by a matrix with entries parameterized by unknown exponents. We derive a best approximate solution of the equation, which yields both vector of coefficients and approximation error parameterized by the exponents. Optimal values of exponents are found by minimization of the approximation error, which is transformed into minimization of a function of exponents over all partitions of a finite set. We solve this minimization problem in terms of max-plus algebra (where addition is defined as maximum and multiplication as arithmetic addition) by using a computational procedure based on the agglomerative clustering technique. This solution is extended to the minimization problem of finding optimal exponents in the polynomial in terms of max-algebra (where addition is defined as maximum). The results obtained are applied to develop new solutions for conventional problems of discrete best Chebyshev approximation of real functions by piecewise linear functions and piecewise Puiseux polynomials. We discuss computational complexity of the proposed solution and estimate upper bounds on the computational time. We demonstrate examples of approximation problems solved in terms of max-plus and max-algebra, and give graphical illustrations. MSC Classification: 15A80 , 90C24 , 41A50 , 41A65 , 65D15

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“…To represent the system of equations at (9) in vector form, we follow the approach in Krivulin (2023) to introduce the following vector-matrix notation:…”

Section: Vector Form Of Approximation Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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