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Some special vapnik-chervonenkis classes

1981
*
Discrete Mathematics
*

For a

doi:10.1016/0012-365x(81)90274-0
fatcat:vd6fssubmfgllmgqwecd7ug5ba
*class*q of subsets of a set X, let V(v) be the smallest n such that no n-element set Fc X has all its subsets of the form ,4 nF, A E V. The condition V(%)C+QC has probabilistic implications. ... Ef any two-element wbset A of X satisfies both A n C = 8 arid A-c b for*some*C, DE. %, then V(q) = 2 if: furd otrly if 0 is linearly -ordered py in&s& Ef S is of the form %Zf(nFS1 Ci: CrE$, i =1,2 ., . ... The study of*special*VCC's has this approacF1 as an underlying motivation.*Some*spe<:ial VCC's have been investigated and their VCN's determined. ...##
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Page 1114 of Mathematical Reviews Vol. , Issue 96b
[page]

1996
*
Mathematical Reviews
*

We then inves- tigate the

*Vapnik*-*Chervonenkis*dimension of certain types of linearly weighted neural networks. ... These algorithms can learn any PAC- learnable*class*and, in*some*cases, settle for significantly less information than the usual labeled examples. ...##
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Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension

1995
*
Journal of combinatorial theory. Series A
*

Let V c {0, 1} n have

doi:10.1016/0097-3165(95)90052-7
fatcat:7ayqaae5h5hclkzazdkhrhvqcm
*Vapnik*-*Chervonenkis*dimension d. Let ~t'(k/n, V) denote the cardinality of the largest W __C V such that any two distinct vectors in W differ on at least k indices. ... We show that /,t'(k/n,V)<_ (cn/(k + d)) d for*some*constant c. This improves on the previous best result of ((cn/k)log(n/k)) d. This new result has applications in the theory of empirical processes. ... In remarks at the end of this paper we consider*some*consequences of Theorem 1 for*some*more general kinds of packing numbers associated with the*Vapnik*-*Chervonenkis*dimension and the pseudodimension, ...##
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On rates of convergence and asymptotic normality in the multiknapsack problem

1991
*
Mathematical programming
*

Note that m6il(n)< 2 11 • The

doi:10.1007/bf01586944
fatcat:uwdmsawaxbbancwpk54jn3mwf4
*class*6D is called a*Vapnik*-*Chervonenkis**class*if m6il(n) < 2n for*some*n;;;iol (cf. [*VAPNIK*& CHERVONENK.IS 1971D. ... A*class*~ of real-valued functions is called a*Vapnik*-*Chervonenkis*graph*class*if the graphs of the functions in ~ form a*Vapnik*-*Chervonenkis*class.The following theorem from [ALEXANDER 1984 ] establishes ...##
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A Lower Bound for Families of Natarajan Dimension d

2001
*
Journal of combinatorial theory. Series A
*

In algorithmic learning theory, the

doi:10.1006/jcta.2000.3160
fatcat:lxcufacqoffrzee2s6l3ceufqu
*Vapnik*-*Chervonenkis*dimension essentially determines the number of samples needed to learn a concept (set) in a given*class*with a given accuracy (see 2] or 1]). ... functions on an n-point set of*Vapnik*-*Chervonenkis*dimension d is P d i=0 ? ... Let us remark that the bipartite graph induced by the considered system on each of the sets fi; jg 3] is the 6-cycle, which is the only extremal K 2;2 -free bipartite graph with*classes*of size 3. ...##
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Comments on "A class of balanced binary sequences with optimal autocorrelation properties" by Lempel, A., et al

1978
*
IEEE Transactions on Information Theory
*

*Vapnik*and A. Ya.

*Chervonenkis*. Our own research was largely completed in 1968 [14] , although we did not submit the manuscript of [15] for publication until 1975. ... As early as 1964,

*Vapnik*and

*Chervonenkis*[3] focused on "the initial choice of the system of acceptable partitions," rather than on specification of classconditional probability distributions (see Section ...

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Page 7668 of Mathematical Reviews Vol. , Issue 95m
[page]

1995
*
Mathematical Reviews
*

{For the entire collection see MR 95d:68001.}
95m:68147 68T05 68Q25 Shinohara, Ayumi (J-K YUS-FI; Fukuoka) Complexity of computing

*Vapnik*-*Chervonenkis*dimension and*some*generalized dimensions. ... Summary: “The*Vapnik*-*Chervonenkis*(VC) dimension is known to be the crucial measure of the polynomial-sample learnability in the PAC-learning model. ...##
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Weak convergence and Glivenko-Cantelli results for empirical processes of u-statistic structure

1989
*
Stochastic Processes and their Applications
*

Empirical processes of U-statistic structure indexed by

doi:10.1016/0304-4149(89)90046-x
fatcat:razuy2cmnnb4pilhr3az3fnry4
*Vapnik*-*Chervonenkis**classes*of sets are studied for independent, but not necessarily identically distributed observations. ... Now we consider*some*countable*Vapnik*-*Chervonenkis**class*% c 0;" 3. ... )*Vapnik*-*Chervonenkis**classes*. ...##
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Author index

1981
*
Discrete Mathematics
*

Dudley,

doi:10.1016/0012-365x(81)90278-8
fatcat:34ez72pakzclxad2o7kykqcei4
*Some**special*'*Vapnik*-*Chervonenkis**classes*(3) 319-321 (1) 79-87 (2) 223-225 ( 1) 89-94 (1) 95-98 (3) 249-258 (2) 197-287 (1) 107-109 (2) 209-221 (3) 313-31s ...##
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Continuous and random Vapnik-Chervonenkis classes

2009
*
Israel Journal of Mathematics
*

In order to do this we generalise the notion of a

doi:10.1007/s11856-009-0094-x
fatcat:4wj47qawvfa7llnbektnaoghdm
*Vapnik*-*Chervonenkis**class*to families of [0,1]-valued functions (a continuous*Vapnik*-*Chervonenkis**class*), and we characterise families of functions having ... Then for*some*r the*class*Q r,r+C 2 /8 is not a*Vapnik*-*Chervonenkis**class*. ... Define the*Vapnik*-*Chervonenkis*index of C, denoted V C(C), to be the minimal d such that f C (d) < 2 d , or infinity if no such d exists. If V C(C) < ∞ then C is a*Vapnik*-*Chervonenkis**class*. ...##
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Page 4431 of Mathematical Reviews Vol. , Issue 87h
[page]

1987
*
Mathematical Reviews
*

C@ is called a

*Vapnik*-*Chervonenkis**class*if dens(C) < oo. C is assumed count- able, or later to satisfy suitable measurability conditions. ... [Remarks on*Vapnik*-*Chervonenkis**classes*and Blei’s combinatorial dimension] Seminar on harmonic analysis, 1983-1984, 92-112, Publ. Math. Orsay, 85-2, Univ. Paris XI, Orsay, 1985. ...##
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Page 1766 of Mathematical Reviews Vol. , Issue 93d
[page]

1993
*
Mathematical Reviews
*

Summary: “We show that a

*class*of subsets of a structure uni- formly definable by a first-order formula is a*Vapnik*-*Chervonenkis**class*if and only if the formula does not have the independence property ... Fuhrken (1-MN) 93d:03039 03C45 03E15 60A10 60F05 Laskowski, Michael C. (1-MSRI)*Vapnik*-*Chervonenkis**classes*of definable sets. J. London Math. Soc. (2) 45 (1992), no. 2, 377-384. ...##
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Page 1288 of Neural Computation Vol. 7, Issue 6
[page]

1995
*
Neural Computation
*

*Some*

*special*

*Vapnik*-

*Chervonenkis*

*classes*. Discrete Math. 33, 313-318. Received March 4, 1994; accepted September 27, 1994. ...

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Book review

1999
*
Automatica
*

The

doi:10.1016/s0005-1098(99)00118-1
fatcat:ykoxqvvemvg5lnprz6pzvsugnq
*Vapnik*}*Chervonenkis*(VC) and Pollard (pseudo-)dimensions are carefully described in Chapter 4, and their relevance to the bounding of certain complexity measures of functional*classes*is described ... This combinatorial dimension essentially measures the size of the largest set of points, for which all the dichotomies may be achieved by functions in the*class*, and was introduced by*Vapnik*and*Chervonenkis*...##
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Guest editor's introduction

1995
*
Machine Learning
*

Building on the work of

doi:10.1007/bf00993407
fatcat:g7mho4pn4bcbbbn5rqh4zfrdqy
*Vapnik*and*Chervonenkis*(*Vapnik*&*Chervonenkis*, 1971) , they showed that the combinatorial parameter of the VC-dimension of a concept*class*essentially characterizes the needed ...*classes*. ...
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