Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection

Eric Bunch, Daniel Dickinson, Jeffery Kline, Glenn Fung

Keywords: magnitude, topology, category theory, machine learning, classification, active learning, outlier detection

TL;DR: The weighting vector used in the definition of the magnitude of a metric space can be used as a boundary detector, and boundaries are useful in machine learning.

Abstract: Metric space magnitude, an active subject of research in algebraic topology, aims to quantify the effective number of distinct points in a 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. 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.

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