Complexity of computing generalized VC-dimensions

Ayumi Shinohara

doi:10.1007/3-540-57868-4_87

Complexity of computing generalized VC-dimensions

Ayumi Shinohara

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Citation (Scopus)

Abstract

In the PAC-learning model, the Vapnik-Chervonenkis (VC) dimension plays the key role to estimate the polynomial-sample learnability of a class of binary functions. For a class of {0,…, m}-valued functions, the notion has been generalized in various ways. This paper investigates the complexity of computing some of generalized VC-dimensions: VC*-dimension, Ψ_*-dimension, and Ψ_G-dimension. For each dimension, we consider a decision problem that is, for a given matrix representing a class F of functions and an integer K, to determine whether the dimension of F is greater than K or not. We prove that the VC*-dimension problem is polynomial-time reducible to the satisfiability problem of length J with O(log²J) variables, which includes the original VC-dimension problem as a special case. We also show that the Ψ_G-dimension problem is still reducible to the satisfiability problem of length J with O(log² J), while the Ψ_*-dimension problem becomes NP-complete.

Original language	English
Title of host publication	Machine Learning
Subtitle of host publication	ECML 1994 - European Conference on Machine Learning, Proceedings
Editors	Francesco Bergadano, Luc De Raedt
Publisher	Springer Verlag
Pages	415-418
Number of pages	4
ISBN (Print)	9783540578680
DOIs	https://doi.org/10.1007/3-540-57868-4_87
Publication status	Published - 1994
Externally published	Yes
Event	European Conference on Machine Learning, ECML 1994 - Catania, Italy Duration: 1994 Apr 6 → 1994 Apr 8

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	784 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	European Conference on Machine Learning, ECML 1994
Country/Territory	Italy
City	Catania
Period	94/4/6 → 94/4/8

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/3-540-57868-4_87

Cite this

Shinohara, A. (1994). Complexity of computing generalized VC-dimensions. In F. Bergadano, & L. De Raedt (Eds.), Machine Learning: ECML 1994 - European Conference on Machine Learning, Proceedings (pp. 415-418). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 784 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-57868-4_87

Complexity of computing generalized VC-dimensions. / Shinohara, Ayumi.

Machine Learning: ECML 1994 - European Conference on Machine Learning, Proceedings. ed. / Francesco Bergadano; Luc De Raedt. Springer Verlag, 1994. p. 415-418 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 784 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Shinohara, A 1994, Complexity of computing generalized VC-dimensions. in F Bergadano & L De Raedt (eds), Machine Learning: ECML 1994 - European Conference on Machine Learning, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 784 LNCS, Springer Verlag, pp. 415-418, European Conference on Machine Learning, ECML 1994, Catania, Italy, 94/4/6. https://doi.org/10.1007/3-540-57868-4_87

Shinohara A. Complexity of computing generalized VC-dimensions. In Bergadano F, De Raedt L, editors, Machine Learning: ECML 1994 - European Conference on Machine Learning, Proceedings. Springer Verlag. 1994. p. 415-418. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/3-540-57868-4_87

Shinohara, Ayumi. / Complexity of computing generalized VC-dimensions. Machine Learning: ECML 1994 - European Conference on Machine Learning, Proceedings. editor / Francesco Bergadano ; Luc De Raedt. Springer Verlag, 1994. pp. 415-418 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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AB - In the PAC-learning model, the Vapnik-Chervonenkis (VC) dimension plays the key role to estimate the polynomial-sample learnability of a class of binary functions. For a class of {0,…, m}-valued functions, the notion has been generalized in various ways. This paper investigates the complexity of computing some of generalized VC-dimensions: VC*-dimension, Ψ*-dimension, and ΨG-dimension. For each dimension, we consider a decision problem that is, for a given matrix representing a class F of functions and an integer K, to determine whether the dimension of F is greater than K or not. We prove that the VC*-dimension problem is polynomial-time reducible to the satisfiability problem of length J with O(log2J) variables, which includes the original VC-dimension problem as a special case. We also show that the ΨG-dimension problem is still reducible to the satisfiability problem of length J with O(log2 J), while the Ψ*-dimension problem becomes NP-complete.

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Complexity of computing generalized VC-dimensions

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