On Frink's Type Metrization of Weighted Graphs
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Published
Feb 20, 2021
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26-37
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Mara Florencia Acosta
Instituto de Matematica Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina.
Hugo Aimar
Instituto de Matematica Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina.
Ivana Gomez
Instituto de Matematica Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina.
Abstract
Using the technique of the metrization theorem of uniformities with countable bases, in this note we provide, test and compare an explicit algorithm to produce a metric d(x, y) between the vertices x and y of an anity weighted undirected graph.
Keywords:
Metrization, uniform spaces, weighted graphs.
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References
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beyond euclidean data. IEEE Signal Processing Magazine. 2017;34(4):18-42.
A mathematical motivation for complex-valued convolutional networks. Neural Computation.;28(5):815-825. PMID: 26890348.
Yann LeCun, Yoshua Bengio, Georey E. Hinton. Deep learning. Nat. 2015;521(7553):436-444.
Karen Simonyan, Andrew Zisserman. Very deep convolutional networks for large-scale image
recognition. In Yoshua Bengio, Yann LeCun, Editors. 3rd International Conference on Learning
Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings; 2015.
Coifman RR, Lafon S, Lee AB, Maggioni M, Nadler B, Warner F, Zucker SW. Geometric diusions as a tool for harmonic analysis and structure denition of data: Diusion maps.
Proceedings of the National Academy of Sciences. 2005;102(21):7426-7431.
Ronald R. Coifman and Stephane Lafon. Diusion maps. Appl. Comput. Harmon. Anal. ;21:5-30.
Frink AH. Distance functions and the metrization problem. Bull. Amer. Math. Soc. ;43(2):133-142.
Chittenden EW. On the metrization problem and related problems in the theory of abstract sets. Bull. Amer. Math. Soc. 1927;33:13-34.
John L. Kelley. General topology. Springer-Verlag. New York-Berlin; 1975. Reprint of the 1955
edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27.
Roberto A. Macas, Carlos Segovia. Lipschitz functions on spaces of homogeneous type. Adv. in Math. 1979;33(3):257-270.
Hugo Aimar, Ivana Gomez. Anity and distance. On the Newtonian structure of some data
kernels. Anal. Geom. Metr. Spaces. 2018;6:89-95.
Bronstein MM, Bruna J, LeCun Y, Szlam A, Vandergheynst P. Geometric deep learning: going
beyond euclidean data. IEEE Signal Processing Magazine. 2017;34(4):18-42.