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Some special vapnik-chervonenkis classes

R.S. Wenocur, R.M. Dudley
1981 Discrete Mathematics  
For a 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.  ... 
doi:10.1016/0012-365x(81)90274-0 fatcat:vd6fssubmfgllmgqwecd7ug5ba

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.  ... 

Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension

David Haussler
1995 Journal of combinatorial theory. Series A  
Let V c {0, 1} n have 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,  ... 
doi:10.1016/0097-3165(95)90052-7 fatcat:7ayqaae5h5hclkzazdkhrhvqcm

On rates of convergence and asymptotic normality in the multiknapsack problem

Sara van de Geer, Leen Stougie
1991 Mathematical programming  
Note that m6il(n)< 2 11 • The 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  ... 
doi:10.1007/bf01586944 fatcat:uwdmsawaxbbancwpk54jn3mwf4

A Lower Bound for Families of Natarajan Dimension d

Paul Fischer, Jiřı́ Matoušek
2001 Journal of combinatorial theory. Series A  
In algorithmic learning theory, the 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.  ... 
doi:10.1006/jcta.2000.3160 fatcat:lxcufacqoffrzee2s6l3ceufqu

Comments on "A class of balanced binary sequences with optimal autocorrelation properties" by Lempel, A., et al

D. Sarwate
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  ... 
doi:10.1109/tit.1978.1055836 fatcat:ba7sae6cjzabvdj74gf56p32am

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.  ... 

Weak convergence and Glivenko-Cantelli results for empirical processes of u-statistic structure

Wilhelm Schneemeier
1989 Stochastic Processes and their Applications  
Empirical processes of U-statistic structure indexed by 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.  ... 
doi:10.1016/0304-4149(89)90046-x fatcat:razuy2cmnnb4pilhr3az3fnry4

Author index

1981 Discrete Mathematics  
Dudley, 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  ... 
doi:10.1016/0012-365x(81)90278-8 fatcat:34ez72pakzclxad2o7kykqcei4

Continuous and random Vapnik-Chervonenkis classes

Itaï Ben Yaacov
2009 Israel Journal of Mathematics  
In order to do this we generalise the notion of a 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.  ... 
doi:10.1007/s11856-009-0094-x fatcat:4wj47qawvfa7llnbektnaoghdm

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.  ... 

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.  ... 

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.  ... 

Book review

1999 Automatica  
The 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  ... 
doi:10.1016/s0005-1098(99)00118-1 fatcat:ykoxqvvemvg5lnprz6pzvsugnq

Guest editor's introduction

Sally A. Goldman
1995 Machine Learning  
Building on the work of 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.  ... 
doi:10.1007/bf00993407 fatcat:g7mho4pn4bcbbbn5rqh4zfrdqy
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