Practical applications of metric space magnitude and weighting vectors

TL;DR

This paper explores the use of metric space magnitude and weighting vectors for boundary detection and machine learning tasks, demonstrating their effectiveness through experiments on benchmark datasets.

Contribution

It introduces novel algorithms based on magnitude and weighting vectors for classification, outlier detection, and active learning, with empirical validation.

Findings

Weighting vectors effectively detect boundaries in Euclidean spaces.

Proposed methods outperform traditional approaches on benchmark datasets.

Magnitude-based algorithms show promise for various machine learning tasks.

Abstract

Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting, the magnitude of a metric space is a real number that aims to quantify the effective number of distinct points in the space. The contribution of each point to a metric space's global magnitude, which is encoded by the {\em weighting vector}, captures much of the underlying geometry of the original metric space. Surprisingly, when the metric space is Euclidean, the weighting vector also serves as an effective tool for boundary detection. This allows the weighting vector to serve as the foundation of novel algorithms for classic machine learning tasks such as classification, outlier detection and active learning. We demonstrate, using experiments and comparisons on classic benchmark datasets, the promise of the proposed magnitude and weighting vector-based approaches.