Book description
Pattern Recognition on Oriented Matroids covers a range of innovative problems in combinatorics, poset and graph theories, optimization, and number theory that constitute a farreaching extension of the arsenal of committee methods in pattern recognition. The groundwork for the modern committee theory was laid in the mid1960s, when it was shown that the familiar notion of solution to a feasible system of linear inequalities has ingenious analogues which can serve as collective solutions to infeasible systems. A hierarchy of dialects in the language of mathematics, for instance, open cones in the context of linear inequality systems, regions of hyperplane arrangements, and maximal covectors (or topes) of oriented matroids, provides an excellent opportunity to take a fresh look at the infeasible system of homogeneous strict linear inequalities – the standard working model for the contradictory twoclass pattern recognition problem in its geometric setting. The universal language of oriented matroid theory considerably simplifies a structural and enumerative analysis of applied aspects of the infeasibility phenomenon.
The present book is devoted to several selected topics in the emerging theory of pattern recognition on oriented matroids: the questions of existence and applicability of matroidal generalizations of committee decision rules and related graphtheoretic constructions to oriented matroids with very weak restrictions on their structural properties; a study (in which, in particular, interesting subsequences of the Farey sequence appear naturally) of the hierarchy of the corresponding tope committees; a description of the threetope committees that are the most attractive approximation to the notion of solution to an infeasible system of linear constraints; an application of convexity in oriented matroids as well as blocker constructions in combinatorial optimization and in poset theory to enumerative problems on tope committees; an attempt to clarify how elementary changes (oneelement reorientations) in an oriented matroid affect the family of its tope committees; a discrete Fourier analysis of the important family of critical tope committees through rank and distance relations in the tope poset and the tope graph; the characterization of a key combinatorial role played by the symmetric cycles in hypercube graphs.
Contents
Oriented Matroids, the Pattern Recognition Problem, and Tope Committees
Boolean Intervals
Dehn–Sommerville Type Relations
Farey Subsequences
Blocking Sets of Set Families, and Absolute Blocking Constructions in Posets
Committees of Set Families, and Relative Blocking Constructions in Posets
Layers of Tope Committees
ThreeTope Committees
Halfspaces, Convex Sets, and Tope Committees
Tope Committees and Reorientations of Oriented Matroids
Topes and Critical Committees
Critical Committees and Distance Signals
Symmetric Cycles in the Hypercube Graphs
Table of contents
 Cover
 Title page
 Copyright
 Dedication
 Contents
 Committees for Pattern Recognition: Infeasible Systems of Linear Inequalities, Hyperplane Arrangements, and Realizable Oriented Matroids
 1 Oriented Matroids, the Pattern Recognition Problem, and Tope Committees
 2 Boolean Intervals
 3 Dehn–Sommerville Type Relations
 4 Farey Subsequences

5 Blocking Sets of Set Families, and Absolute Blocking Constructions in Posets
 5.1 Blocking Elements and Complementing Elements of Subposets
 5.2 Blocking Elements in Direct Products of Posets
 5.3 The Blocker Map and the Complementary Map on Antichains
 5.4 The Lattice of Blockers
 5.5 Deletion and Contraction
 5.6 The Blocker, Deletion, Contraction, and Maps on Posets
 5.7 The Blocker, Deletion, Contraction, Powers of 2, and the SelfDual Clutters
 5.8 The (X, k)Blocker Map
 5.9 (X, k)Deletion and (X, k)Contraction
 Notes

6 Committees of Set Families, and Relative Blocking Constructions in Posets
 6.1 Relatively Blocking Elements of Antichains
 6.2 Absolutely Blocking Elements of Antichains, and Convex Subposets
 6.3 A Connection Between Absolute and Relative Blocking Constructions
 6.4 The Structure of the Subposets of Relatively Blocking Elements, and Their Enumeration
 6.5 Principal Order Ideals and Farey Subsequences
 6.6 Relatively Blocking Elements in Graded Posets, and Farey Subsequences
 6.7 Relatively Blocking Elements in the Boolean Lattices
 6.8 Relatively Blocking Elements b with the Property b ∧ −b = Ô in the Boolean Lattices of Subsets of the Sets ±{1, . . ., m}
 6.9 Relatively Blocking Elements in the Posets Isomorphic to the Face Lattices of Crosspolytopes
 6.10 Relatively Blocking Elements in the Principal Order Ideals of Binomial Posets
 Notes
 7 Layers of Tope Committees
 8 ThreeTope Committees
 9 Halfspaces, Convex Sets, and Tope Committees
 10 Tope Committees and Reorientations of Oriented Matroids
 11 Topes and Critical Committees
 12 Critical Committees and Distance Signals
 13 Symmetric Cycles in the Hypercube Graphs
 Bibliography
 List of Notation
 Index
Product information
 Title: Pattern Recognition on Oriented Matroids
 Author(s):
 Release date: September 2017
 Publisher(s): De Gruyter
 ISBN: 9783110530841
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