Pattern Recognition on Oriented Matroids: Critical Committees and Distance Signals

TL;DR

This paper explores the relationship between critical committees in oriented matroids and the Fourier transform of distance signals derived from symmetric cycles in the tope graph, revealing new insights into their structure.

Contribution

It introduces a novel connection between critical committees in oriented matroids and the Fourier analysis of associated distance signals.

Findings

Number of committee members relates to Fourier components of the distance signal.

Establishes a link between combinatorial structures and signal processing techniques.

Provides a new method to analyze oriented matroids using spectral analysis.

Abstract

If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the positive tope, is a critical committee for M that votes for the base tope. We consider the sequence zR of poset ranks of the elements from the vertex sequence of R as a fragment of a signal with period 2t and relate the number of members of the committee min V(R) to the magnitudes of [t/2] components, with odd indices, of the discrete Fourier transform of the distance vector zR.