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	VC dimension - Wikipedia, the free encyclopedia
		The VC dimension (for Vapnik Chervonenkis dimension) is a measure of the capacity of a statistical classification algorithm.
		The VC dimension of a model f is the maximum h such that some data point set of cardinality h can be shattered by f.
		The VC dimension has utility in statistical learning theory, because it can predict a probabilistic upper bound on the test error of a classification model.
	en.wikipedia.org /wiki/VC_dimension (382 words)

	dimension
		Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space.
		For any topological space, the Lebesgue covering dimension is defined to be n if any open cover has a refinement (a second cover where each element is a subset of an element in the first cover) such that no point is included in more than n+1 elements.
		The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.fact-library.com /dimension.html (624 words)

	[No title]
		This was related to the VC dimension and to the MAD of graphs.
		The VC dimension of graphs was investigated earlier by Anthony, Brightwell, Cooper, and Kranakis.
		Focusing on the VC dimension and its applications in the light of Graph Theory produces structural results mainly through characterization problems and computational questions as the speed of computing dimensions.
	www.clarkson.edu /honors/research/papers/Shehu-Amarda.doc (556 words)

	VC dimension - TheBestLinks.com - Algorithm, Cardinality, Degree, Polynomial, ... (Site not responding. Last check: 2007-09-20)
		VC dimension, Algorithm, Cardinality, Degree, Polynomial, Upper bound...
		The VC dimension of a model $f is the maximum h such that some data point set of cardinality h can be shattered by f .$
		with probability $1-\eta, where h is the$ VC dimension of the classification model, and $N is the size of the training set.$
	www.thebestlinks.com /VC_dimension.html (418 words)

	Encyclopedia: VC theory
		Vapnik Chervonenkis theory (also known as VC theory) was developed during 1960-1990 by Vladimir Vapnik and Alexey Chervonenkis.
		VC theory is also referred to as statistical learning theory by Vapnik and his close colleagues.
		VC theory contains important concepts such as the VC dimension and structural risk minimization.
	www.nationmaster.com /encyclopedia/VC-theory (258 words)

	Support Vector Machines
		"The VC dimension is a property of a set of functions {f(α)} (again, we use α as a generic set of parameters: a choice of α specifies a particular function), and can be defined for various classes of function f.
		The VC dimension for the set of functions {f(α)} is defined as the maximum number of training points that can be shattered by {f(α)}.
		Despite the fact that the VC dimension is very important, the unfortunate reality is that its analytic estimations can be used only for the simplest sets of functions.
	svms.org /vc-dimension (817 words)

	ipedia.com: VC dimension Article (Site not responding. Last check: 2007-09-20)
		The VC dimension is a measure of the capacity of a classification algorithm.
		The VC dimension of a model is the maximum such that some data point set of cardinality can be shattered by.
		with probability, where is the VC dimension of the classification model, and is the size of the training set.
	www.ipedia.com /vc_dimension.html (411 words)

	Ruhr-Uni Bochum - Lehrstuhl Mathematik und Informatik - VC Dimension Bounds for Higher-Order Neurons
		We calculate upper and lower bounds on the Vapnik-Chervonenkis dimension and the pseudo dimension for higher-order neurons that allow unrestricted interactions among the input variables.
		In particular, we show that the degree of interaction is irrelevant for the VC dimension and that the individual degree of the variables plays only a minor role.
		Further, our results reveal that the crucial parameters that affect the VC dimension of higher-order neurons are the input dimension and the maximum number of occurrences of each variable.
	www.ruhr-uni-bochum.de /lmi/mschmitt/vchigher-abs.html (149 words)

	[No title]
		Keywords: linear systems identification, learning theory, VC dimension 1 Introduction The problem of systems identification may be seen as an instance of the general question of "learning" an unknown function.
		In terms of n (the dimension of the state space) and k (the band-width) the upper bound is of the form O(n3 log 2 (nk)).
		Note that if the system dimension n is small compared to the band-width k the VC-dimension upper and lower bounds in Theorems 3 and 4 become tighter, both being of the form c log 2 k (with different values of the constant c).
	www.math.rutgers.edu /~sontag/vcdim-ctlinsys.html (8058 words)

	VC dimension and VC confidence
		VC confidence is defined as the second term on the right hand side of Eqn.
		VC dimension can be defined for various classes of learning machines.
		In [Vapnik, 1995], it is stressed that the VC dimension and hence the capacity does not depend on the number of free parameters directly, i.e.
	svr-www.eng.cam.ac.uk /~kkc21/thesis_main/node11.html (259 words)

	Support Vector Machines - The Book
		VC theory has since been used to analyse the performance of learning systems as diverse as decision trees, neural networks, and others; many learning heuristics and principles used in practical applications of machine learning have been explained in terms of VC theory.
		VC theory has recently also come to be known as Statistical Learning Theory, and is extensively described in the recent book of Vapnik, and in other books that preceded it, as well as earlier papers by Vapnik and Chervonenkis.
		The reason why margin analysis requires different tools from VC theory is that the quantity used to characterise the richness of a hypothesis class, the margin, depends on the data.
	www.support-vector.com /chapter_4.html (1015 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		The notion of embedding a class of dichotomies in a class of linear half spaces is central to the support vector machines paradigm.
		We examine the question of determining the minimal Euclidean dimension and the maximal margin that can be obtained when the embedded class has a finite VC dimension.
		We show that an overwhelming majority of the family of finite concept classes of any constant VC dimension cannot be embedded in low-dimensional half spaces.
	jmlr.csail.mit.edu /papers/v3/bendavid02a.html (204 words)

	Exact VC-Dimension of Boolean Monomials (ResearchIndex) (Site not responding. Last check: 2007-09-20)
		The paper provides the exact values for the VC dimension of Boolean monomials.
		Abstract: We show that the Vapnik-Chervonenkis dimension of Boolean monomials over n variables is at most n for all n 2.
		It follows that the VC-dimension is determined exactly and is, except for n = 1, equal to the VC-dimension of the proper subclass of monotone monomials.
	citeseer.ist.psu.edu /88000.html (292 words)

	96-56: Vapnik-Chervonenkis Dimension of Recurrent Neural Networks (Site not responding. Last check: 2007-09-20)
		This paper provides lower and upper bounds for the VC dimension of such networks.
		In contrast, for feedforward networks, VC dimension bounds can be expressed as a function of $w$ only.
		\item For the standard sigmoid $\sigma(x)=1/(1+e^{-x})$, the VC dimension is between $\nopar\leninp$ and %PK: $\nopar^4\leninp^2$.
	dimacs.rutgers.edu /TechnicalReports/abstracts/1996/96-56.html (217 words)

	VC Dimension (cont.) (Site not responding. Last check: 2007-09-20)
		The VC dimension of {f(?)} is the maximum number of training points that can be shattered by {f(?)}
		For example, the VC dimension of a set of oriented lines in R2 is three.
		In general, the VC dimension of a set of oriented hyperplanes in Rn is n+1.
	www.stat.rutgers.edu /~madigan/datamining/svm/sld005.htm (99 words)

	RHUL Computer Science: Research: Computational Learning (Site not responding. Last check: 2007-09-20)
		If the VC dimension is low, the expected probability of error is low as well, which means good generalisation.
		The generalization ability of this learning machine depends on the VC dimension of the set of functions that the machine implements rather than on the dimensionality of the space.
		A function that describes the data well and belongs to a set with low VC dimension will generalize well regardless of the dimensionality of the space.
	www.dcs.rhbnc.ac.uk /research/compint/areas/comp_learn/sv/index.shtml (169 words)

	NeuroCOLT:Adaptive and Self-Confident Online Learning Algorithms
		We establish superlinear lower bounds on the Vapnik Chervonenkis (VC) dimension of neural networks with one hidden layer and local receptive field neurons.
		As the main result we show that every reasonably sized standard network of radial basis function (RBF) neurons has VC dimension $\Omega(W\log k)$, where $W$ is the number of parameters and $k$ the number of nodes.
		In particular, they imply that the pseudo dimension and the fat-shattering dimension of these networks is superlinear as well, and they yield lower bounds even when the input dimension is fixed.
	www.neurocolt.com /abs/2001/abs01105.html (178 words)

	Computational Learning Theory
		The maximum number of dichotomies induced by hypotheses in H on any set of m instances is defined as the growth function of H with respect to m.
		The Vapnik-Chervonenkis dimension (VCdim) of H is the largest m such that the corresponding growth function is equal to 2^{m}.
		That is, H can induce all possible dichotomies of m instances drawn from the instance space if and only if the Vapnik-Chervonenkis dimension of H is m.
	www.cise.ufl.edu /~fu/Lecture/Learn/theory-fu.html (471 words)

	On the VC dimension of bounded margin classifiers - Hush, Scovel (ResearchIndex) (Site not responding. Last check: 2007-09-20)
		On the VC dimension of bounded margin classifiers (2000)
		Hush, D., and Scovel, C., On the VC dimension of bounded margin classifiers, submitted to Machine Learning, June, 1999.
		On the VC dimension of bounded margin classifiers - Hush, Scovel (1999)
	citeseer.lcs.mit.edu /hush00vc.html (428 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		The special issue arose out of an ICMS Workshop on the VC Dimension though submission is not restricted to those who attended the workshop.
		For more information on the Vapnik-Chervonenkis dimension and the aims of the workshop and hence also the Special Issue, please consult the `scientific aims' available through the homepage:
		Applications of the VC Dimension in Complexity Theory
	compgeom.cs.uiuc.edu /~jeffe/compgeom/files/VC.html (199 words)

	SVM training and SRM
		, the VC dimension of a nonlinear SVM is N +1, where N is the dimensionality of space
		[Vapnik, 1995] stated that the VC dimension is bounded by the inequality in Eqn.
		actually gives the VC dimension of a gap-tolerant classifier but not the VC dimension of a SVM classifier.
	svr-www.eng.cam.ac.uk /~kkc21/thesis_main/node15.html (229 words)

	Neural Computation - On the Practical Applicability of VC Dimension Bounds - The MIT Press
		Specifically, we are interested in bounds on the sample complexity for the problem of training a pattern classifier such that we can expect it to perform valid generalization.
		Early results using the VC dimension, while being extremely powerful, suffered from the fact that their sample complexity predictions were rather impractical.
		The results of these experiments indicate that the more recent theories provide sample complexity predictions that are significantly more applicable in practice than those provided by earlier theories; however, we also find that the recent theories still have significant shortcomings.
	mitpress.mit.edu /catalog/item?sid=8ACF50D4-4362-4484-9FDD-C9F4124557DC&ttype=6&tid=1839 (189 words)

	World History :: Encyclopedia Index -- Vc (Site not responding. Last check: 2007-09-20)
		INDEX OF ARTICLES: Vc Articles are indexed by the first word of the title, including "A," "The," etc.
		VCs of the First World War - The Western Front 1915
		VCs of the First World War: Arras and Messines 1917
	www.worldhistory.com /wiki/Vc.htm (140 words)

	Caltech Computer Science Technical Reports - Invariance Hints and the VC Dimension
		Fyfe, William John Andrew (1992) Invariance Hints and the VC Dimension.
		The goal is to substitute examples of the invariant for examples of the function; the extent to which we can actually do this depends on the appropriate VC dimensions.
		You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format.
	caltechcstr.library.caltech.edu /87 (252 words)

	Published in Neural Computation (6)5, 1994, pp 851-876
		If we assume that the constants obtained are universal, we can then use the function obtained this way to measure the capacity of other learning machines.
		Intuitively, an appropriate weight decay term will make these dimensions effectively "invisible" to the linear classifier, thereby reducing the measured effective VC-dimension.
		The Vapnik-Chervonenkis dimension is a mathematical quantity that measures the "capacity" of a learning machine.
	www.djvuzone.org /djvu/sci/yann/vc/index.emb.html (1250 words)

	VC Dimension Bounds for Product Unit Networks (Site not responding. Last check: 2007-09-20)
		We establish bounds on the Vapnik-Chervonenkis (V-C) dimension that can be used to assess the generalization capabilities of these networks.
		In particular, we show that the VC dimension for these networks is not larger than the best known bound for sigmoidal networks.
		For higher-order networks, we derive upper bounds that are independent of the degree of these networks.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/ijcnn/2000/0619/04/0619toc.xml&DOI=10.1109/IJCNN.2000.860767 (184 words)

	IngentaConnect On the complexity of approximating the VC dimension (Site not responding. Last check: 2007-09-20)
		IngentaConnect On the complexity of approximating the VC dimension
		We study the complexity of approximating the VC dimension of a collection of sets, when the sets are encoded succinctly by a small circuit.
		We prove analogous results for theq-ary VC dimension, where the approximation threshold is q.
	www.ingentaconnect.com /content/ap/ss/2002/00000065/00000004/art00022 (169 words)

	VC++ - How to create a 3 dimension array dynamicly
		want In the case where all of the dimensions are variable, you cannot use
		Another idea is to use a class to handle everything for you.
		This class can remember to dimensions so that it can be deleted correctly.
	www.codecomments.com /archive292-2004-5-198095.html (805 words)

	NeuroCOLT: Neural Networks and Computational Learning Theory
		It is assumed the reader has some familiarity with neural networks, but otherwise the survey is largely self-contained.
		The basic PAC model of concept learning is discussed and the key results involving the Vapnik-Chervonenkis dimension are derived.
		Implications for the theory of artificial neural networks are discussed through a survey of known results on the VC-dimension of neural nets.
	www.neurocolt.com /abs/1994/abs94003.html (163 words)

	A VC: A New Dimension?
		Because I believe that relevancy is the next dimension and stored user preferences and data are the foundation of delivering it.
		Cookies are one of the ways to do that and I think they are a very important part of the next big thing.
		Fred Wilson is in the want of a new dimension: Outlook is for calendaring, e-mail and contacts, Word for writing..., appropriately termed as islands/ghettos.
	avc.blogs.com /a_vc/2005/08/a_new_dimension.html (2488 words)

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