Optimal expected-distance separating halfspace

One recently proposed criterion to separate two datasets in discriminant analysis, is to use a hyperplane which minimises the sum of distances to it from all the misclassified data points. Here all distances are supposed to be measured by way of some fixed norm, while misclassification means lying o...

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Detalles Bibliográficos
Autores: Carrizosa Priego, Emilio José, Plastria, Frank
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2008
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/44827
Acceso en línea:http://hdl.handle.net/11441/44827
https://doi.org/10.1287/moor.1070.030
Access Level:acceso abierto
Palabra clave:Gauge-distance to hyperplane
Separating halfspace
Discriminant analysis
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spelling Optimal expected-distance separating halfspaceCarrizosa Priego, Emilio JoséPlastria, FrankGauge-distance to hyperplaneSeparating halfspaceDiscriminant analysisOne recently proposed criterion to separate two datasets in discriminant analysis, is to use a hyperplane which minimises the sum of distances to it from all the misclassified data points. Here all distances are supposed to be measured by way of some fixed norm, while misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study the problem of determining such an optimal halfspace when points are distributed according to an arbitrary random vector X in Rd,. In the unconstrained case in dimension d, we prove that any optimal separating halfspace always balances the misclassified points. Moreover, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d affinely independent points. It follows that the problem is polynomially solvable in fixed dimension by an algorithm of O(n d+1) when the support of X consists of n points. All these results are strengthened in the one-dimensional case, yielding an algorithm with complexity linear in the cardinality of the support of X. If a different norm is used for each data set in order to measure distances to the hyperplane, or if all distances are measured by a fixed gauge, the balancing property still holds, and we show that, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d − 1 affinely independent data points. These results extend in a natural way when we allow constraints modeling that certain points are forced to be correctly classified.Ministerio de Ciencia y TecnologíaINFORMS (Institute for Operations Research and Management Sciences)Estadística e Investigación OperativaFQM329: OptimizacionMinisterio de Ciencia y Tecnología (MCYT). España2008info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/44827https://doi.org/10.1287/moor.1070.030reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésMathematics of Operations Research, 33 (3), 662-677.BFM2002-04525-C02-02BFM2002-11282-Ehttp://pubsonline.informs.org/doi/pdf/10.1287/moor.1070.0309info:eu-repo/semantics/openAccessoai:idus.us.es:11441/448272026-06-17T12:51:07Z
dc.title.none.fl_str_mv Optimal expected-distance separating halfspace
title Optimal expected-distance separating halfspace
spellingShingle Optimal expected-distance separating halfspace
Carrizosa Priego, Emilio José
Gauge-distance to hyperplane
Separating halfspace
Discriminant analysis
title_short Optimal expected-distance separating halfspace
title_full Optimal expected-distance separating halfspace
title_fullStr Optimal expected-distance separating halfspace
title_full_unstemmed Optimal expected-distance separating halfspace
title_sort Optimal expected-distance separating halfspace
dc.creator.none.fl_str_mv Carrizosa Priego, Emilio José
Plastria, Frank
author Carrizosa Priego, Emilio José
author_facet Carrizosa Priego, Emilio José
Plastria, Frank
author_role author
author2 Plastria, Frank
author2_role author
dc.contributor.none.fl_str_mv Estadística e Investigación Operativa
FQM329: Optimizacion
Ministerio de Ciencia y Tecnología (MCYT). España
dc.subject.none.fl_str_mv Gauge-distance to hyperplane
Separating halfspace
Discriminant analysis
topic Gauge-distance to hyperplane
Separating halfspace
Discriminant analysis
description One recently proposed criterion to separate two datasets in discriminant analysis, is to use a hyperplane which minimises the sum of distances to it from all the misclassified data points. Here all distances are supposed to be measured by way of some fixed norm, while misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study the problem of determining such an optimal halfspace when points are distributed according to an arbitrary random vector X in Rd,. In the unconstrained case in dimension d, we prove that any optimal separating halfspace always balances the misclassified points. Moreover, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d affinely independent points. It follows that the problem is polynomially solvable in fixed dimension by an algorithm of O(n d+1) when the support of X consists of n points. All these results are strengthened in the one-dimensional case, yielding an algorithm with complexity linear in the cardinality of the support of X. If a different norm is used for each data set in order to measure distances to the hyperplane, or if all distances are measured by a fixed gauge, the balancing property still holds, and we show that, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d − 1 affinely independent data points. These results extend in a natural way when we allow constraints modeling that certain points are forced to be correctly classified.
publishDate 2008
dc.date.none.fl_str_mv 2008
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/44827
https://doi.org/10.1287/moor.1070.030
url http://hdl.handle.net/11441/44827
https://doi.org/10.1287/moor.1070.030
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Mathematics of Operations Research, 33 (3), 662-677.
BFM2002-04525-C02-02
BFM2002-11282-E
http://pubsonline.informs.org/doi/pdf/10.1287/moor.1070.0309
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv INFORMS (Institute for Operations Research and Management Sciences)
publisher.none.fl_str_mv INFORMS (Institute for Operations Research and Management Sciences)
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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