Graphs for Pattern Recognition: Infeasible Systems of Linear Inequalities

Graphs for Pattern Recognition: Infeasible Systems of Linear Inequalities

by Damir Gainanov

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Overview

This monograph deals with mathematical constructions that are foundational in such an important area of data mining as pattern recognition. By using combinatorial and graph theoretic techniques, a closer look is taken at infeasible systems of linear inequalities, whose generalized solutions act as building blocks of geometric decision rules for pattern recognition.Infeasible systems of linear inequalities prove to be a key object in pattern recognition problems described in geometric terms thanks to the committee method. Such infeasible systems of inequalities represent an important special subclass of infeasible systems of constraints with a monotonicity property – systems whose multi-indices of feasible subsystems form abstract simplicial complexes (independence systems), which are fundamental objects of combinatorial topology.The methods of data mining and machine learning discussed in this monograph form the foundation of technologies like big data and deep learning, which play a growing role in many areas of human-technology interaction and help to find solutions, better solutions and excellent solutions. Contents:PrefacePattern recognition, infeasible systems of linear inequalities, and graphsInfeasible monotone systems of constraintsComplexes, (hyper)graphs, and inequality systemsPolytopes, positive bases, and inequality systemsMonotone Boolean functions, complexes, graphs, and inequality systemsInequality systems, committees, (hyper)graphs, and alternative coversBibliographyList of notationIndex

Product Details

ISBN-13: 9783110480139
Publisher: De Gruyter
Publication date: 10/10/2016
Pages: 158
Product dimensions: 6.69(w) x 9.45(h) x (d)
Age Range: 18 Years

About the Author

Damir Gainanov, Ural Federal University, Russia.

Table of Contents

Preface v

Pattern recognition, infeasible systems of linear inequalities, and graphs 1

1 Infeasible monotone systems of constraints 7

1.1 Structural and combinatorial properties of infeasible monotone systems of constraints 8

1.2 Abstract simplicial complexes and monotone Boolean functions 12

Notes 18

2 Complexes, (hyper)graphs, and inequality systems 20

2.1 The graph of an independence system 20

2.2 The hypergraph of an independence system 31

2.3 The graph of maximal feasible subsystems of an infeasible system of linear inequalities 33

2.4 The hypergraph of maximal feasible subsystems of an infeasible system of linear inequalities 53

Notes 56

3 Polytopes, positive bases, and inequality systems 58

3.1 Faces and diagonals of convex polytopes 58

3.2 Positive bases of linear spaces 67

3.3 Polytopes and infeasible systems of inequalities 75

Notes 90

4 Monotone Boolean functions, complexes, graphs, and inequality systems 93

4.1 Optimal inference of monotone Boolean functions 93

4.2 An inference algorithm for monotone Boolean functions associated with graphs 100

4.3 Monotone Boolean functions and inequality systems 110

Notes 112

5 Inequality systems, committees, (hyper)graphs, and alternative covers 115

5.1 The graph of MFSs of an infeasible system of linear inequalities and committees 116

5.2 The hypergraph of MF5s of an infeasible system of linear inequalities and committees 125

5.3 Alternative covers 126

Notes 130

Bibliography 133

List of notation 141

Index 144

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