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Metric space magnitude, an active field of research in algebraic topology, isa scalar quantity that summarizes the effective number of distinct points thatlive in a general metric space. The {\em weighting vector} is a closely-relatedconcept that captures, in a nontrivial way, much of the underlying geometry ofthe original metric space. Recent work has demonstrated that when the metricspace is Euclidean, the weighting vector serves as an effective tool forboundary detection. We recast this result and show the weighting vector may beviewed as a solution to a kernelized SVM. As one consequence, we apply this newinsight to the task of outlier detection, and we demonstrate performance thatis competitive or exceeds performance of state-of-the-art techniques onbenchmark data sets. Under mild assumptions, we show the weighting vector,which has computational cost of matrix inversion, can be efficientlyapproximated in linear time. We show how nearest neighbor methods canapproximate solutions to the minimization problems defined by SVMs.

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