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

Metric space magnitude, an active field of research in algebraic topology, is\na scalar quantity that summarizes the effective number of distinct points that\nlive in a general metric space. The {\\em weighting vector} is a closely-related\nconcept that captures, in a nontrivial way, much of the underlying geometry of\nthe original metric space. Recent work has demonstrated that when the metric\nspace is Euclidean, the weighting vector serves as an effective tool for\nboundary detection. We recast this result and show the weighting vector may be\nviewed as a solution to a kernelized SVM. As one consequence, we apply this new\ninsight to the task of outlier detection, and we demonstrate performance that\nis competitive or exceeds performance of state-of-the-art techniques on\nbenchmark data sets. Under mild assumptions, we show the weighting vector,\nwhich has computational cost of matrix inversion, can be efficiently\napproximated in linear time. We show how nearest neighbor methods can\napproximate solutions to the minimization problems defined by SVMs.\n