Gauge groups and data classification. (Q1406139)

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.