Abstract
The earliest published instance found of combinatorial quantities later occurring in the work of Vapnik and Chervonenkis was in a paper of Jakob Steiner in 1826. The next, still more pertinent occurrence found was in work of Ludwig Schläfli done around 1850 but not published until 1901, after his death. The nineteenth century work was on subsets of Euclidean spaces cut by intersections of finitely many half-spaces. Then there is another long gap until a paper of T.M. Cover, who cited Schläfli, in 1965, preceding by a few years the landmark announcement by Vapnik and Chervonenkis in 1968 and longer paper of 1971. Further history is given about Steiner, Schläfli, and some of their contemporary mathematicians and about the initial reception of the Vapnik and Chervonenkis work.
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Notes
- 1.
It has always been (and still is) customary at the Institute of Control Sciences to list authors’ names in alphabetical order. See also a similar discussion in Chap. 5 (Eds.).
References
Assouad, P.: Densité et dimension. Ann. de l’Institut Fourier (Grenoble) 33, 233–282 (1983)
Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Classifying learnable geometric concepts with the Vapnik–Chervonenkis dimension. In: Proceedings of the 18th Annual Symposium on the Theory of Computing, pp. 273–282. ACM (1986)
*Burckhardt, J.J.: Ludwig Schläfli. Birkhäuser, Basel (1948)
Burckhardt, J.J.: Schläfli, Ludwig. In: Complete Dictionary of Scientific Biography. Charles Scribner’s Sons (2008). http://www.encyclopedia.com/doc/1G2-2830903877.html. Earlier Burckhardt wrote a longer biography (1948, [3]) in German
Cover, T.M.: Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Transactions on Electronic Computers EC-14, 326–334 (1965)
Coxeter, H.S.M.: Regular Polytopes. Methuen, London (1948). Reprinted in 1949 (Pitman, New York). Second edition: 1963 (Macmillan, New York). Reprinted in 1964 (Dover, New York). Third edition: 1973 (Dover, New York)
Fejes Tóth, L.: Regular Figures. Pergamon, London (1964)
*Grassmann, H.: Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics, in German). Wiegand, Leipzig (1844). Second edition: 1862
Grattan-Guinness, I. (ed.): Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Routledge, London (1994). Two volumes, total 1806 pp.; Johns Hopkins University Press, 2003
Grünbaum, B.: Regular polyhedra. In: [9], chap. 7 (§7.1), pp. 866–876. Routledge (1994)
*Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discret. Comput. Geom. 2, 127–151 (1987)
*Kirkman, T.P.: On a problem in combinatorics. Camb. Dublin Math. J. II, 191–204 (1847)
O’Connor, J.J., Robertson, E.F.: August Leopold Crelle. http://www-history.mcs.st-and.ac.uk/Biographies/Crelle.html (2000)
O’Connor, J.J., Robertson, E.F.: Ludwig Schläfli. http://www-history.mcs.st-andrews.ac.uk/Biographies/Schlafli.html (2007)
O’Connor, J.J., Robertson, E.F.: Jakob Steiner. http://www-history.mcs.st-andrews.ac.uk/Biographies/Steiner.html (2009)
Sauer, N.: On the density of families of sets. Journal of Combinatorial Theory, Series A 13, 145–147 (1972)
Schläfli, L.: Gesammelte Schriften (Collected Works) (1901). Republished as “Gesammelte mathematische Abhandlungen” in 1950–1956 by Birkhäuser, Basel
Schläfli, L.: Theorie der vielfachen Kontinuität (Theory of Multidimensional Continua, in German). Denkschriften der Schweizerischen Naturforschenden Gesellschaft (Memoirs of the Swiss Scientific Society). J. H. Graf, Bern (1901). Republished by Cornell University Library, 1991. Also included in [17]
Schlesinger, M.I., Hlaváč, V.: Ten Lectures on Statistical and Structural Pattern Recognition. Kluwer, Dordrecht (2002)
Scholz, E.: Topology: geometric, algebraic. In: [9], chap. 7 (§7.10), pp. 927–938. Routledge (1994)
Steele, J.M.: Combinatorial entropy and uniform limit laws. Ph.D. dissertation, Mathematics, Stanford University (1975)
Steele, J.M.: Empirical discrepancies and subadditive processes. Ann. Probab. 6, 118–127 (1978)
Steiner, J.: Einige Gesetze über die Theilung der Ebene und des Raumes (Some theorems on the division of plane and space, in German). Journal für die Reine und Angewandte Mathematik 1, 349–364 (1826)
*Steiner, J.: Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander... (Systematic development of the dependence of geometric objects on each other..., in German). Fincke, Berlin (1832). Available as e-book (Barnes and Noble)
*Steiner, J.: Combinatorische Aufgabe (A combinatorial problem, in German). Journal für die Reine und Angewandte Mathematik 45, 181–182 (1853)
Vapnik, V.N.: Statistical Learning Theory. Wiley, New York (1998)
Vapnik, V.N., Chervonenkis, A.Y.: On the uniform convergence of the frequencies of occurrence of events to their probabilities. Dokl. Akad. Nauk SSSR 181, 781–783 (1968). Sov. Math. Dokl. 9, 915–918
Vapnik, V.N., Chervonenkis, A.Y.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–279 (Russian), 264–280 (English) (1971). This volume, Chap. 3
Vapnik, V.N., Chervonenkis, A.Y.: Теория распознавания образов: статистические проблемы обучения (Theory of Pattern Recognition: Statistical Problems of Learning; in Russian). Nauka, Moscow (1974)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press (1922, republished in 1966)
Wilson, R.J., Lloyd, E.K.: Combinatorics. In: [9], chap. 7 (§7.13), pp. 952–965. Routledge (1994)
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Dudley, R.M. (2015). Sketched History: VC Combinatorics, 1826 up to 1975. In: Vovk, V., Papadopoulos, H., Gammerman, A. (eds) Measures of Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-21852-6_4
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