Abstract
Given M vectors in N-dimensional attribute space, it is much easier to find M hyperplanes that separate each of the vectors from all the others than to solve M arbitrary linear dichotomies with approximately equal class memberships. An explanation of the rapid growth with M and N of the number of separable one-against-all linear halfplane dichotomies is proposed in terms of convex polyhedra in a hyperspherical shell. The counterintuitive surge is illustrated by averaged results on pseudo-random integer arrays obtained by Linear Programming and Neural Networks. Although the initial motivation arose from seemingly arbitrary rankings of scientists and universities, this project is not directed at any application.
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Nagy, G., Krishnamoorthy, M. (2022). One-Against-All Halfplane Dichotomies. In: Krzyzak, A., Suen, C.Y., Torsello, A., Nobile, N. (eds) Structural, Syntactic, and Statistical Pattern Recognition. S+SSPR 2022. Lecture Notes in Computer Science, vol 13813. Springer, Cham. https://doi.org/10.1007/978-3-031-23028-8_19
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