Solution algorithms for fuzzy relational equations with max-product composition

Bourke, Mary M.; Fisher, D. Grant

doi:10.1016/s0165-0114(96)00246-1

Cited by 96 publications

(52 citation statements)

References 31 publications

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“…This is a partially ordered set that is determined (Bourke and Fisher 1998;Di Nola and Lettieri 1989;Markovskii 2005) by the minimal solutions and by the unique maximum solution, bearing in mind the density of ordering also. The property density of ordering, see Di Nola and Lettieri (1989), Proposition 4…”

Section: Inverse Problemmentioning

confidence: 99%

“…Universal algorithm and software for solving max − min and min − max fuzzy relational equations is proposed in Peeva (2002Peeva ( , 2006 and Peeva and Kyosev (2004). Concerning fuzzy linear system of equations with max-product composition (Di Nola and Lettieri 1989;, the results concern greatest solution (Bourke and Fisher 1998), minimal solutions (in some references procedures pretend to yield to minimal solutions, but in fact they yield to some non-minimal solutions as well), estimating time complexity of the problem, applications in optimization problems (Guu and Wu 2002;Fang 1999, 2001;Loetamonphong et al 2002). The relationship with the covering problem is subject of Markovskii (2005), where two methods for solving such fuzzy linear systems (algebraic and with table decomposition) are discussed and an algorithm is proposed, realizing table decomposition method.…”

Section: Introductionmentioning

confidence: 99%

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Algorithm for Solving Max-product Fuzzy Relational Equations

Peeva

Kyosev

2006

Soft Comput

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Section: Inverse Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algorithm for Solving Max-product Fuzzy Relational Equations

Peeva

Kyosev

2006

Soft Comput

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“…They generalize calculus of relations [8]. Without any exaggeration one may say that they can be used as a formal framework for numerous pursuits carried out in the setting of fuzzy sets, starting from various schemes of approximate reasoning (including a compositional rule of inference), models of decision-making, and relational structures in data mining [1,2,6,7,9,10,19]. A generic version of the equation arises in the form…”

mentioning

confidence: 99%

“…Generally, from the practical standpoint we are concerned with a collection of fuzzy relational equations (relational constraints) rather than a single equation. Then (1) generalizes to the form Yk Ak Bk Ck Á Á Á Xk R 2 with k 1; 2; . .…”

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Modularization of fuzzy relational equations

Pedrycz

Vasilakos

2002

Soft Computing - A Fusion of Foundations, Methodologies and App

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In this study, we are concerned with a modularization of fuzzy relational equations that is converting a highly dimensional (multivariable) relational equation into a series of single input fuzzy relational equations. The problem originates from a need of handling (estimating) highly dimensional relational structures and is inherently associated with the curse of dimensionality present in relational fuzzy models. We propose a twolayer architecture and discuss a detailed optimization scheme leading to the determination of the fuzzy relations occurring there. Illustrative numerical studies are also included.1 Introduction: coping with multivariable fuzzy relation equations Fuzzy relational equations have been around for several decades almost from the inception of fuzzy sets [3, 4, 5, 11±18]. They generalize calculus of relations [8]. Without any exaggeration one may say that they can be used as a formal framework for numerous pursuits carried out in the setting of fuzzy sets, starting from various schemes of approximate reasoning (including a compositional rule of inference), models of decision-making, and relational structures in data mining [1,2,6,7,9,10,19]. A generic version of the equation arises in the formIn the above expression, A; B; C; . . . ; X are input fuzzy sets, R becomes a multidimensional fuzzy relation and Y denotes an output fuzzy set. The above topology of the equation can be viewed as a generic relational model of a multivariable static system. The convolution of the input fuzzy sets with the fuzzy relation can be realized in many possible ways; the s±t composition is a general option to work with while the well-known max±min composition comes as a special case. Generally, from the practical standpoint we are concerned with a collection of fuzzy relational equations (relational constraints) rather than a single equation. Then (1) generalizes to the form Yk Ak Bk Ck Á Á Á Xk R 2 with k 1; 2; . . . ; N.While the relational calculus supported by (1) or (2) exhibits a number of appealing features and is quite general, there are still open practical questions. They concern a way of solving such fuzzy relational equation(s). The approaches towards handling these equations fall under two general categories. First, analytical solutions hold under assumptions (e.g., a type of the convolution operation; in general we cannot come up with a general way of solving equations for the s±t composition). Second, if we con®ne ourselves to optimization methods (usually in the form of some iterative algorithms), it is very likely that these could maintain their effectiveness for relatively small dimensionality of the problem. What really hampers relational equations is the dimensionality of the problem that is cast in such setting. The fuzzy relation required to handle (1) grows up in size very quickly. This immediately raises an issue of ef®cient storage of the fuzzy relation itself. When proceeding with numeric optimization (estimation of R through some iterative computing), large fuzzy relations jeopardize the...

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Quaternionic Fuzzy Neural Network for View‐Invariant Color Face Image Recognition

Wong

Lee

Loo

et al. 2013

Complex‐Valued Neural Networks

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This paper presents an effective color image processing system view-invariant person face image recognition for Max Planck Institute Kybernetik (MPIK) dataset. The proposed system can recognize face images of view-invariant person by correlating the input face images with the reference face image and classifying them according to the correct persons' name/ID indeed. It has been carried out by constructing a complex quaternion correlator and a max-product fuzzy neural network classifier. Two classification parameters, namely discrete quaternion correlator output (p-value) and the peak to sidelobe ratio (PSR), were used in classifying the input face images, and to categorise them either into the authentic class or non-authentic class. Besides, a new parameter called G-value is also introduced in the proposed view-invariant color face image recognition system for better classification purpose.Experimental results shows that the proposed view-invariant color face image recognition system outperforms the conventional NMF, BDNMF and hypercomplex Gabor filter in terms of consumption of enrollment time, recognition time and accuracy in classifying MPIK color face images which are viewinvariant, noise influenced and scale invariant. Povzetek: Predstavljena je metoda prepoznavanja obrazov, testirana na domeni Max Planck Institute Kybernetik.

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Solution algorithms for fuzzy relational equations with max-product composition

Cited by 96 publications

References 31 publications

Algorithm for Solving Max-product Fuzzy Relational Equations

Algorithm for Solving Max-product Fuzzy Relational Equations

Modularization of fuzzy relational equations

Quaternionic Fuzzy Neural Network for View‐Invariant Color Face Image Recognition

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