Gauge groups and data classification.
type: Item
instance of: scholarly article
title: Gauge groups and data classification.
zbMATH Open document ID: 1033.65009
DOI: 10.1016/S0096-3003(02)00115-7
author: D. Massart
published in: Applied Mathematics and Computation
publication date: 2003-09-09T00:00:00Z
review text: The author presents an algorithm to find a separating surface for two sets of points \(P\) and \(Q\) in \({\mathbb R}^d\). The method is based on the gauge action of the group \(SO(d)\) on \({\mathbb R^d}\). Given a separating surface \(\Omega(x)=0\) which separates \(P\) and \(Q\) via \(\Omega(x)>0\) for \(x\in \widetilde P\) and \(\Omega(x)<0\) for \(x\in\widetilde Q\), where \(\widetilde P\subset P\) and \(\widetilde Q\subset Q\), the idea is to reduce the number of misclassifications (i.e., the size of the sets \(P\setminus\widetilde P\) and \(Q\setminus \widetilde Q\)) by a transformation of the surface \(\Omega(x)=0\). For each vector in \({\mathbb R}^d\) there is an associated antisymmetric \(d\times d\) matrix which parameterizes a subset of the gauged group \(SO(d)\). The action of the Lie group \(SO(d)\) in \({\mathbb R}^d\) defines a diffeomorphism on \({\mathbb R}^d\). The new surface \(\Omega'(x)=0\) is the image of the surface \(\Omega(x)=0\) under this diffeomorphism. As an application of the correctness proof for the algorithm, the author shows that every dichotomy \((P,Q)\) in \( {\mathbb R}^d\) can be separated by a polynomial surface.
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 68T10
zbMATH DE Number: 1978002
zbMATH Keywords: pattern recognition
zbMATH Keywords: dichotomy
zbMATH Keywords: gauge classification
zbMATH Keywords: Kirchberger's theorem
zbMATH Keywords: algorithm
zbMATH Keywords: separating surface
reviewed by: Wolfgang zu Castell
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1978002