Community detection, pattern recognition, and hypergraph-based learning: approaches using metric geometry and persistent homology
This work addresses the problem of analyzing higher-order relations in complex data for researchers in machine learning and data science, but it appears incremental as it builds on existing topological and nearest neighbors approaches.
The paper tackles community detection and pattern recognition in hypergraph data by introducing a new topological structure resembling a metric space, and proposes modified nearest neighbors methods for discrete structures, applying them to sign prediction problems.
Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One of the main contributions of our paper is to introduce a new topological structure to hypergraph data which bears a resemblance to a usual metric space structure. Using this new topological space structure of hypergraph data, we propose several approaches to study community detection problem, detecting persistent features arising from homological structure of hypergraph data. Also based on the topological space structure of hypergraph data introduced in our paper, we introduce a modified nearest neighbors methods which is a generalization of the classical nearest neighbors methods from machine learning. Our modified nearest neighbors methods have an advantage of being very flexible and applicable even for discrete structures as in hypergraphs. We then apply our modified nearest neighbors methods to study sign prediction problem in hypegraph data constructed using our method.