On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions

Akito Sakurai

doi:10.1007/3-540-57370-4_52

On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions

Akito Sakurai

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

2 Citations (Scopus)

Abstract

We consider the problem of determining VC-dimension (Formula Found) of depth four n-input 1-output threshold circuits with h elements. Best known asymptotic lower bounds and upper bounds are proved, that is, when (Formula Found) is upper bounded by ((h²/3)+nh(log h)(1+o(1)) and lower bounded by (1/2)((h²/4)+nh)(log h)(1 − o(1)). We also consider the problem of determining complexity c₃(N) of Boolean-valued functions defined on N-pointsets in Rⁿ, measured by the number of threshold elements, with which we can construct a depth four circuit to realize the functions. We also show the best known upper and lower bounds, that is, when N → ∞, the complexity is upper bounded by (Formula Found) and lower bounded by (Formula Found)

Original language	English
Title of host publication	Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings
Editors	Klaus P. Jantke, Shigenobu Kobayashi, Etsuji Tomita, Takashi Yokomori
Publisher	Springer Verlag
Pages	251-264
Number of pages	14
ISBN (Print)	9783540573708
DOIs	https://doi.org/10.1007/3-540-57370-4_52
Publication status	Published - 1993
Externally published	Yes
Event	4th Workshop on Algorithmic Learning Theory, ALT 1993 - Tokyo, Japan Duration: 1993 Nov 8 → 1993 Nov 10

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	744 LNAI
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	4th Workshop on Algorithmic Learning Theory, ALT 1993
Country/Territory	Japan
City	Tokyo
Period	93/11/8 → 93/11/10

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/3-540-57370-4_52

Cite this

Sakurai, A. (1993). On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. In K. P. Jantke, S. Kobayashi, E. Tomita, & T. Yokomori (Eds.), Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings (pp. 251-264). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 744 LNAI). Springer Verlag. https://doi.org/10.1007/3-540-57370-4_52

On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. / Sakurai, Akito.

Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. ed. / Klaus P. Jantke; Shigenobu Kobayashi; Etsuji Tomita; Takashi Yokomori. Springer Verlag, 1993. p. 251-264 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 744 LNAI).

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

Sakurai, A 1993, On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. in KP Jantke, S Kobayashi, E Tomita & T Yokomori (eds), Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 744 LNAI, Springer Verlag, pp. 251-264, 4th Workshop on Algorithmic Learning Theory, ALT 1993, Tokyo, Japan, 93/11/8. https://doi.org/10.1007/3-540-57370-4_52

Sakurai A. On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. In Jantke KP, Kobayashi S, Tomita E, Yokomori T, editors, Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. Springer Verlag. 1993. p. 251-264. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-57370-4_52

Sakurai, Akito. / On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings. editor / Klaus P. Jantke ; Shigenobu Kobayashi ; Etsuji Tomita ; Takashi Yokomori. Springer Verlag, 1993. pp. 251-264 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions

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