Witnessing and guiding sets of tangles

TL;DR

This paper studies witnessing and guiding sets of tangles in discrete data, providing bounds on their sizes and characterizations that improve understanding of cluster-like structures in complex datasets.

Contribution

It introduces bounds on witnessing set sizes for tangles and characterizes tangles with reliable guiding functions, advancing the theoretical framework of tangles in data analysis.

Findings

Witnessing sets of tangles have sizes bounded exponentially in k.

Improved bounds over previous results by Grohe and Schweizer.

Characterization of tangles with reliable guiding functions.

Abstract

Tangles o er a way to indirectly but precisely capture cluster-like though possibly fuzzy substructures in discrete data. In this paper, we analyze witnessing and guiding sets of tangles that can help to find proper cluster candidates for given tangles. We show that every k-tangle has a witnessing set whose size is bounded in an exponential function in k which improves a result of Grohe and Schweizer. Further, we generalize a result of Diestel, Elbracht and Jacobs by providing a characterization of tangles that have a guiding function of some given reliability.