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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

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CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86


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2299-3274
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Affinity and Distance. On the Newtonian Structure of Some Data Kernels

Hugo Aimar
  • 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
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  • 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
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Published Online: 2018-06-15 | DOI: https://doi.org/10.1515/agms-2018-0005

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

Keywords : uniform spaces; metric spaces; affinity kernel

References

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About the article

Received: 2017-08-15

Revised: 2018-04-06

Accepted: 2018-04-23

Published Online: 2018-06-15


Citation Information: Analysis and Geometry in Metric Spaces, Volume 6, Issue 1, Pages 89–95, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2018-0005.

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© 2018 Ivana Gómez, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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