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

AIMAR, HUGO; GÓMEZ, IVANA

Analysis and Geometry in Metric Spaces

De Gruyter

Año: 2018 vol. 6 p. 89 - 95

Let X be aset. Let K(x, y) > 0 be a measure of the affinity between the data points xand 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 onX under two mild conditions on K. The first is that the affinity of each x toitself is infinite and that for x ≠ y the affinity is positive and finite. Thesecond is a quantitative transitivity; if the affinity between x and y islarger than λ > 0and the affinity of y and z is also larger than λ, then the affinity between x and z is largerthan ν(λ). The function ν is concave, increasing, continuousfrom R+ onto R+ with ν(λ) < λ for every λ > 0.