A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks

A sufficient condition that a region be classifiable by a two-layer feedforward neural net (a two-layer perceptron) using threshold activation functions is that either it be a convex polytope or that intersected with the complement of a convex polytope in its interior, or that intersected with the c...

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Detalles Bibliográficos
Autores principales:	Cabrelli, Carlos Alberto, Molter, Ursula Maria
Publicado:	2000
Materias:	Convex recursive deletion region Neural network Two-layer perceptron Algorithms Backpropagation Computational geometry Matrix algebra Multilayer neural networks Optimization Theorem proving Vectors Two layer perceptron Feedforward neural networks
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10459227_v11_n3_p811_Cabrelli http://hdl.handle.net/20.500.12110/paper_10459227_v11_n3_p811_Cabrelli
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_10459227_v11_n3_p811_Cabrelli
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spelling	paper:paper_10459227_v11_n3_p811_Cabrelli2023-06-08T16:01:11Z A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks Cabrelli, Carlos Alberto Molter, Ursula Maria Convex recursive deletion region Neural network Two-layer perceptron Algorithms Backpropagation Computational geometry Matrix algebra Multilayer neural networks Optimization Theorem proving Vectors Convex recursive deletion region Two layer perceptron Feedforward neural networks A sufficient condition that a region be classifiable by a two-layer feedforward neural net (a two-layer perceptron) using threshold activation functions is that either it be a convex polytope or that intersected with the complement of a convex polytope in its interior, or that intersected with the complement of a convex polytope in its interior or . . . recursively. These have been called convex recursive deletion (CoRD) regions. We give a simple algorithm for finding the weights and thresholds in both layers for a feedforward net that implements such a region. The results of this work help in understanding the relationship between the decision region of a perceptron and its corresponding geometry in input space. Our construction extends in a simple way to the case that the decision region is the disjoint union of CoRD regions (requiring three layers). Therefore this work also helps in understanding how many neurons are needed in the second layer of a general three-layer network. In the event that the decision region of a network is known and is the union of CoRD regions, our results enable the calculation of the weights and thresholds of the implementing network directly and rapidly without the need for thousands of backpropagation iterations. © 2000 IEEE. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2000 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10459227_v11_n3_p811_Cabrelli http://hdl.handle.net/20.500.12110/paper_10459227_v11_n3_p811_Cabrelli
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Convex recursive deletion region Neural network Two-layer perceptron Algorithms Backpropagation Computational geometry Matrix algebra Multilayer neural networks Optimization Theorem proving Vectors Convex recursive deletion region Two layer perceptron Feedforward neural networks
spellingShingle	Convex recursive deletion region Neural network Two-layer perceptron Algorithms Backpropagation Computational geometry Matrix algebra Multilayer neural networks Optimization Theorem proving Vectors Convex recursive deletion region Two layer perceptron Feedforward neural networks Cabrelli, Carlos Alberto Molter, Ursula Maria A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks
topic_facet	Convex recursive deletion region Neural network Two-layer perceptron Algorithms Backpropagation Computational geometry Matrix algebra Multilayer neural networks Optimization Theorem proving Vectors Convex recursive deletion region Two layer perceptron Feedforward neural networks
description	A sufficient condition that a region be classifiable by a two-layer feedforward neural net (a two-layer perceptron) using threshold activation functions is that either it be a convex polytope or that intersected with the complement of a convex polytope in its interior, or that intersected with the complement of a convex polytope in its interior or . . . recursively. These have been called convex recursive deletion (CoRD) regions. We give a simple algorithm for finding the weights and thresholds in both layers for a feedforward net that implements such a region. The results of this work help in understanding the relationship between the decision region of a perceptron and its corresponding geometry in input space. Our construction extends in a simple way to the case that the decision region is the disjoint union of CoRD regions (requiring three layers). Therefore this work also helps in understanding how many neurons are needed in the second layer of a general three-layer network. In the event that the decision region of a network is known and is the union of CoRD regions, our results enable the calculation of the weights and thresholds of the implementing network directly and rapidly without the need for thousands of backpropagation iterations. © 2000 IEEE.
author	Cabrelli, Carlos Alberto Molter, Ursula Maria
author_facet	Cabrelli, Carlos Alberto Molter, Ursula Maria
author_sort	Cabrelli, Carlos Alberto
title	A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks
title_short	A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks
title_full	A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks
title_fullStr	A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks
title_full_unstemmed	A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks
title_sort	constructive algorithm to solve "convex recursive deletion" (cord) classification problems via two-layer perceptron networks
publishDate	2000
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10459227_v11_n3_p811_Cabrelli http://hdl.handle.net/20.500.12110/paper_10459227_v11_n3_p811_Cabrelli
work_keys_str_mv	AT cabrellicarlosalberto aconstructivealgorithmtosolveconvexrecursivedeletioncordclassificationproblemsviatwolayerperceptronnetworks AT molterursulamaria aconstructivealgorithmtosolveconvexrecursivedeletioncordclassificationproblemsviatwolayerperceptronnetworks AT cabrellicarlosalberto constructivealgorithmtosolveconvexrecursivedeletioncordclassificationproblemsviatwolayerperceptronnetworks AT molterursulamaria constructivealgorithmtosolveconvexrecursivedeletioncordclassificationproblemsviatwolayerperceptronnetworks
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A constructive algorithm to solve "Convex Recursive Deletion" (CoRD) classification problems via two-layer perceptron networks

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