TY - GEN

T1 - Complexity of computing generalized VC-dimensions

AU - Shinohara, Ayumi

PY - 1994

Y1 - 1994

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=0347816633&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347816633&partnerID=8YFLogxK

U2 - 10.1007/3-540-57868-4_87

DO - 10.1007/3-540-57868-4_87

M3 - Conference contribution

AN - SCOPUS:0347816633

SN - 9783540578680

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 415

EP - 418

BT - Machine Learning

A2 - Bergadano, Francesco

A2 - De Raedt, Luc

PB - Springer Verlag

T2 - European Conference on Machine Learning, ECML 1994

Y2 - 6 April 1994 through 8 April 1994

ER -