Pattern Recognition on Oriented Matroids - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Pattern Recognition on Oriented Matroids write by Andrey O. Matveev. This book was released on 2017-09-11. Pattern Recognition on Oriented Matroids available in PDF, EPUB and Kindle. Pattern Recognition on Oriented Matroids covers a range of innovative problems in combinatorics, poset and graph theories, optimization, and number theory that constitute a far-reaching extension of the arsenal of committee methods in pattern recognition. The groundwork for the modern committee theory was laid in the mid-1960s, 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 two-class 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 graph-theoretic 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 three-tope 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 (one-element 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 Three-Tope 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
Pattern Recognition on Oriented Matroids
Pattern Recognition on Oriented Matroids - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Pattern Recognition on Oriented Matroids write by Andrey O. Matveev. This book was released on 2017-09-11. Pattern Recognition on Oriented Matroids available in PDF, EPUB and Kindle. Pattern Recognition on Oriented Matroids covers a range of innovative problems in combinatorics, poset and graph theories, optimization, and number theory that constitute a far-reaching extension of the arsenal of committee methods in pattern recognition. The groundwork for the modern committee theory was laid in the mid-1960s, 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 two-class 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 graph-theoretic 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 three-tope 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 (one-element 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 Three-Tope 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
Computational Oriented Matroids
Computational Oriented Matroids - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Computational Oriented Matroids write by Jürgen Bokowski. This book was released on 2006-05-08. Computational Oriented Matroids available in PDF, EPUB and Kindle. Oriented matroids play the role of matrices in discrete geometry, when metrical properties, such as angles or distances, are neither required nor available. Thus they are of great use in such areas as graph theory, combinatorial optimization and convex geometry. The variety of applications corresponds to the variety of ways they can be defined. Each of these definitions corresponds to a differing data structure for an oriented matroid, and handling them requires computational support, best realised through a functional language. Haskell is used here, and, for the benefit of readers, the book includes a primer on it. The combination of concrete applications and computation, the profusion of illustrations, many in colour, and the large number of examples and exercises make this an ideal introductory text on the subject. It will also be valuable for self-study for mathematicians and computer scientists working in discrete and computational geometry.
Pattern Recognition
Pattern Recognition - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Pattern Recognition write by Jürgen Beyerer. This book was released on 2017-12-04. Pattern Recognition available in PDF, EPUB and Kindle. The book offers a thorough introduction to Pattern Recognition aimed at master and advanced bachelor students of engineering and the natural sciences. Besides classification - the heart of Pattern Recognition - special emphasis is put on features, their typology, their properties and their systematic construction. Additionally, general principles that govern Pattern Recognition are illustrated and explained in a comprehensible way. Rather than presenting a complete overview over the rapidly evolving field, the book is to clarifies the concepts so that the reader can easily understand the underlying ideas and the rationale behind the methods. For this purpose, the mathematical treatment of Pattern Recognition is pushed so far that the mechanisms of action become clear and visible, but not farther. Therefore, not all derivations are driven into the last mathematical detail, as a mathematician would expect it. Ideas of proofs are presented instead of complete proofs. From the authors’ point of view, this concept allows to teach the essential ideas of Pattern Recognition with sufficient depth within a relatively lean book. Mathematical methods explained thoroughly Extremely practical approach with many examples Based on over ten years lecture at Karlsruhe Institute of Technology For students but also for practitioners
Oriented Matroids
Oriented Matroids - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Oriented Matroids write by Anders Björner. This book was released on 1999-11-18. Oriented Matroids available in PDF, EPUB and Kindle. First comprehensive, accessible account; second edition has expanded bibliography and a new appendix surveying recent research.
