Singular 0/1-Matrices, and the Hyperplanes Spanned by Random 0/1-Vectors
Combinatorics, Probability and Computing
Remote sensing image segmentation by active queries
Pattern Recognition
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We consider hyperplanes spanned by vertices of the unit $d$-cube. We classify these hyperplanes by parallelism to coordinate axes, by symmetry of the $d$-cube vertices they avoid, as well as by so-called hull-honesty. (Hull-honest hyperplanes are those whose intersection figure with the $d$-cube coincides with the convex hull of the $d$-cube vertices they contain; they do not cut $d$-cube edges properly.) We describe relationships between these classes and give the exact number of hull-honest hyperplanes in general dimensions. An experimental enumeration of all spanned hyperplanes up to dimension eight showed us the intrinsic difficulty of developing a general enumeration scheme. Motivation for considering such hyperplanes stems from coding theory, from linear programming, and from the theory of machine learning.