Affinity and Distance. On the Newtonian Structure of Some Data Kernels

  • 1 Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ. CCT-CONICET-Santa Fe, Predio “Dr. Alberto Cassano”, Colectora Ruta Nac. 168 km 0, Paraje El Pozo, S3007ABA , Santa Fe, Argentina

Abstract

Let X be a set. Let K(x, y) > 0 be a measure of the affinity between the data points x and y. We prove that K has the structure of a Newtonian potential K(x, y) = φ(d(x, y)) with φ decreasing and d a quasi-metric on X under two mild conditions on K. The first is that the affinity of each x to itself is infinite and that for x ≠ y the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between x and y is larger than λ > 0 and the affinity of y and z is also larger than λ, then the affinity between x and z is larger than ν(λ). The function ν is concave, increasing, continuous from R+ onto R+ with ν(λ) < λ for every λ > 0

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Analysis and Geometry in Metric Spaces (AGMS) is a fully peer-reviewed, open access electronic journal that publishes cutting-edge original research on analytical and geometrical problems in metric spaces and their mathematical applications. It features articles making connections among relevant topics in this field.

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