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Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection
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Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection
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Metric space magnitude, an active field of research in algebraic topology, is a scalar quantity that summarizes the effective number of distinct points that live in a general metric space. The {\em weighting vector} is a closely-related concept that captures, in a nontrivial way, much of the underlying geometry of the original metric space. Recent work has demonstrated that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection. We recast this result and show the weighting vector may be viewed as a solution to a kernelized SVM. As one consequence, we apply this new insight to the task of outlier detection, and we demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets. Under mild assumptions, we show the weighting vector, which has computational cost of matrix inversion, can be efficiently approximated in linear time. We show how nearest neighbor methods can approximate solutions to the minimization problems defined by SVMs.
Forward citations
Cited by 4 Pith papers
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The microscopic weighting on a metric space
Every finite metric space of strictly negative type admits a canonical signed measure (the microscopic weighting) that maximizes distance energy and equals the derivative of the magnitude function at zero
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A Centrality Measure Using Magnitude Homology
A new family of graph centrality measures based on the change in (Eulerian) magnitude homology after deleting a vertex, with a proven locality property.
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Magnitude homology and Euler characteristics of directed acyclic graphs
A decategorification shortcut turns magnitude-homology Euler characteristic computation for DAGs into a polynomial-arithmetic linear solve, with a proof-of-concept on MLP activation graphs.
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Scalably computing metric magnitude
Hierarchical low-rank solvers beat dense and sparsified approaches for metric magnitude solves in experiments up to n=30,000, with a projected path to n≈10^5 via a containerized STRUMPACK/MPI pipeline.
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