Pattern recognition on oriented matroids

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This work explores innovative problems in combinatorics, poset and graph theories, optimization, and number theory, significantly extending committee methods in pattern recognition. The foundation of modern committee theory was established in the mid-1960s, revealing analogues of solutions to feasible linear inequalities that can address infeasible systems. A hierarchy of mathematical dialects, including open cones and maximal covectors of oriented matroids, offers a fresh perspective on the infeasible system of homogeneous strict linear inequalities, a model for the contradictory two-class pattern recognition problem. The universal language of oriented matroid theory simplifies the structural and enumerative analysis of this phenomenon. The book delves into various topics within the theory of pattern recognition on oriented matroids, such as the existence and applicability of matroidal generalizations of committee decision rules, the study of tope committees, and the description of three-tope committees as approximations to solutions of infeasible linear constraints. It also covers the application of convexity in oriented matroids, the impact of one-element reorientations on tope committees, and a discrete Fourier analysis of critical tope committees. Additionally, it characterizes the combinatorial role of symmetric cycles in hypercube graphs, enriching the understanding of these mathematical constructs.