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Research Article

Extension of semi-Lipschitz maps on non-subadditive quasi-metric spaces: new tools for Artificial Intelligence

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Pages 123-146 | Received 13 Sep 2022, Published online: 14 Apr 2023

References

  • F. Albiac, The role of local convexity in Lipschitz maps, J. Convex Anal. 18(4) (2011), 983–997.
  • T.V. An, L.Q. Tuyen, and N.V. Dung, Stone-type theorem on b-metric spaces and applications, Topology and its Applications 185 (2015), 50–64. doi: 10.1016/j.topol.2015.02.005
  • C. Anil, J. Lucas, and R. Grosse, Sorting out Lipschitz function approximation, Proceedings of the 36th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 97, pp. 291–301, Microtome Publishing, Brookline, MA, 2019.
  • A.V. Arutyunov and A.V. Greshnov, Theory of (q1, q2)-quasimetric spaces and coincidence points, Dokl. Math. 94 (2016), 434–437. doi: 10.1134/S1064562416040232
  • J.M. Calabuig, H. Falciani, and E.A. Sánchez-Pérez, Dreaming machine learning: Lipschitz extensions for reinforcement learning on financial markets, Neurocomputing 398 (2020), 172–184. doi: 10.1016/j.neucom.2020.02.052
  • S. Cobzaş, Functional Analysis in Asymmetric Normed Spaces, Birkhäuser, Basel, 2013.
  • S. Cobzaş and S. Czerwik, The completion of generalized b-metric spaces and fixed points, Fixed Point Theory 21(1) (2020), 133–150. doi: 10.24193/fpt-ro.2020.1.10
  • S. Cobzaş, R. Miculescu, and A. Nicolae, Lipschitz Functions, Springer, Cham, 2019.
  • J.-P. Doignon and J.-Cl. Falmagne, Spaces for the assessment of knowledge, International Journal of Man-Machine Studies 23 (1985), 175–196. doi: 10.1016/S0020-7373(85)80031-6
  • R. Fagin and L. Stockmeyer, Relaxing the triangle inequality in pattern matching, International Journal of Computer Vision 30 (1998), 219–231. doi: 10.1023/A:1008023416823
  • H. Falciani and E.A. Sánchez-Pérez, Semi-Lipschitz functions and machine learning for discrete dynamical systems on graphs, Mach. Learn. 111 (2022), 1765–1797. doi: 10.1007/s10994-022-06130-x
  • L. Gottlieb, A. Kontorovich, and R. Krauthgamer, Efficient Regression in Metric Spaces via Approximate Lipschitz Extension, IEEE Transactions on Information Theory 63(8) (2017), 4838–4849. doi: 10.1109/TIT.2017.2713820
  • N.J. Kalton, N.T. Peck, and J.W. Roberts, An F-space Sampler, London Mathematical Society Lecture Note Series, Vol. 89, Cambridge University Press, Cambridge, 1984.
  • W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014.
  • R. Kopperman, All topologies come from generalized metrics, The American Mathematical Monthly 95(2) (1988), 89–97. doi: 10.1080/00029890.1988.11971974
  • D. Kurepa, Tableaux ramifiés d’ensembles, espaces pseudo-distanciés, C. R. Acad. Sci. Paris 198 (1934), 1563–1565.
  • R. Kyng, A. Rao, S. Sachdeva, and D.A. Spielman, Algorithms for Lipschitz Learning on Graphs, Proceedings of The 28th Conference on Learning Theory, Proceedings of Machine Learning Research, Vol. 40, pp. 1190–1223, Microtome Publishing, Brookline, MA, 2015.
  • R.A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Advances in Mathematics 33 (1979), 257–270. doi: 10.1016/0001-8708(79)90012-4
  • A.S. Mashhour, A.A. Allam, F.S. Mahmoud, and F.H. Khedr, On supratopological spaces, Indian J. Pure Appl. Math. 14(4) (1983), 502–510.
  • E.J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40(12) (1934), 837–842. doi: 10.1090/S0002-9904-1934-05978-0
  • C. Mustăţa, Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx. 30(1) (2001), 61–67. doi: 10.33993/jnaat301-682
  • M. Paluszyński and K. Stempak, On quasi-metric and metric spaces, Proc. Amer. Math. Soc. 137(12) (2009), 4307–4312. doi: 10.1090/S0002-9939-09-10058-8
  • S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, Journal of Approximation Theory 103(2) (2000), 292–301. doi: 10.1006/jath.1999.3439
  • B. Samet, C. Vetro, and F. Vetro, From metric spaces to partial metric spaces, Fixed Point Theory and Applications 2013(5) (2013), 1–11.
  • S. Tiefenbrun, US foreign trade zones, tax-free trade zones of the world, and their impact on the US economy, Journal of International Business and Law 12(2) (2014), 149–222.
  • W. Usino, A.S. Prabuwono, K.H.S. Allehaibi, A. Bramantoro, A. Hasniaty, and W. Amaldi, Document similarity detection using k-means and cosine distance, International Journal of Advanced Computer Science and Applications 10(2) (2019), 165–170. doi: 10.14569/IJACSA.2019.0100222
  • U. von Luxburg and O. Bousquet, Distance-Based Classification with Lipschitz Functions, J. Mach. Learn. Res. 512(1) (2004), 669–695.
  • W.A. Wilson, On semi-metric spaces, American Journal of Mathematics 53(2) (1931), 361–373. doi: 10.2307/2370790
  • W.A. Wilson, On quasi-metric spaces, American Journal of Mathematics 53(3) (1931), 675–684. doi: 10.2307/2371174
  • H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36(1) (1934), 63–89. doi: 10.1090/S0002-9947-1934-1501735-3
  • G. Wu, H. Lin, E. Fu, and L. Wang, An Improved K-means Algorithm for Document Clustering, 2015 International Conference on Computer Science and Mechanical Automation, pp. 65–69, IEEE, New York, 2015.
  • Q. Xia, The geodesic problem in quasimetric spaces, J. Geom. Anal. 19 (2009), 452–479. doi: 10.1007/s12220-008-9065-4

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