Korean Journal of Mathematics

  • 제32권1호
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  • Pages.43-57
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  • 2024
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  • 1976-8605(pISSN)
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  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
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VC-DIMENSION AND DISTANCE CHAINS IN 𝔽dq

  • ;
  • Ruben Ascoli (Department of Mathematics, Georgia Institute of Technology) ;
  • Livia Betti (Department of Mathematics, University of Rochester) ;
  • Justin Cheigh (Department of Mathematics, Williams College) ;
  • Alex Iosevich (Department of Mathematics, University of Rochester) ;
  • Ryan Jeong (Department of Mathematics, University of Cambridge) ;
  • Xuyan Liu (Department of Mathematics, Brown University) ;
  • Brian McDonald (Department of Mathematics, University of Georgia) ;
  • Wyatt Milgrim (Department of Mathematics, University of California Los Angeles) ;
  • Steven J. Miller (Department of Mathematics, Williams College) ;
  • Francisco Romero Acosta (Department of Mathematics, Virginia Polytechnic Institute and State University) ;
  • Santiago Velazquez Iannuzzelli (Department of Mathematics, Northwestern University)
  • 투고 : 2022.11.26
  • 심사 : 2023.06.22
  • 발행 : 2024.03.30
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초록

Given a domain X and a collection H of functions h : X → {0, 1}, the Vapnik-Chervonenkis (VC) dimension of H measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions ℋ'2t(E) : 𝔽2q → {0, 1}, corresponding to indicator functions of circles centered at points in a subset E ⊆ 𝔽2q. They showed that when |E| is large enough, the VC-dimension of ℋ'2t(E) is the same as in the case that E = 𝔽2q. We study a related hypothesis class, ℋdt(E), corresponding to intersections of spheres in 𝔽dq, and ask how large E ⊆ 𝔽dq needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever |E| ≥ Cdqd-1/(d-1) for d ≥ 3, the VC-dimension of ℋdt(E) is as large as possible. We get a slightly stronger result if d = 3: this result holds as long as |E| ≥ C3q7/3. Furthermore, when d = 2 the result holds when |E| ≥ C2q7/4.

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참고문헌

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