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VC Dimension

by Martin Sewell , 2008
"... VC dimension (for Vapnik Chervonenkis dimension) measures the capacity of a hypothesis space. Capacity is a measure of complexity and measures the expressive power, richness or flexibility of a set of functions by assessing how wiggly its members can be. The definitions below are taken from Vapnik ( ..."
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VC dimension (for Vapnik Chervonenkis dimension) measures the capacity of a hypothesis space. Capacity is a measure of complexity and measures the expressive power, richness or flexibility of a set of functions by assessing how wiggly its members can be. The definitions below are taken from Vapnik

VC-Dimension of Rule Sets

by Olcay Taner Yıldız
"... Abstract—In this paper, we give and prove lower bounds of the VC-dimension of the rule set hypothesis class where the input features are binary or continuous. The VC-dimension of the rule set depends on the VC-dimension values of its rules and the number of inputs. Index Terms—VC-Dimension, Rule set ..."
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Abstract—In this paper, we give and prove lower bounds of the VC-dimension of the rule set hypothesis class where the input features are binary or continuous. The VC-dimension of the rule set depends on the VC-dimension values of its rules and the number of inputs. Index Terms—VC-Dimension, Rule

Classification with infinite VC-dimension

by O Teytaud
"... Families of finite VC-dimension are not able of ensuring a good asymptotic behavior for any distribution. We sum up and compare paradigms for working without finiteness of the VC-dimension and compare them. ..."
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Families of finite VC-dimension are not able of ensuring a good asymptotic behavior for any distribution. We sum up and compare paradigms for working without finiteness of the VC-dimension and compare them.

VC Dimension of Neural Networks

by Eduardo D. Sontag - Neural Networks and Machine Learning , 1998
"... . This paper presents a brief introduction to Vapnik-Chervonenkis (VC) dimension, a quantity which characterizes the difficulty of distribution-independent learning. The paper establishes various elementary results, and discusses how to estimate the VC dimension in several examples of interest in ne ..."
Abstract - Cited by 24 (3 self) - Add to MetaCart
. This paper presents a brief introduction to Vapnik-Chervonenkis (VC) dimension, a quantity which characterizes the difficulty of distribution-independent learning. The paper establishes various elementary results, and discusses how to estimate the VC dimension in several examples of interest

VC-dimension of Exterior Visibility

by Volkan Isler, Sampath Kannan, Kostas Daniilidis, Pavel Valtr - IEEE Trans. Pattern Analysis and Machine Intelligence , 2004
"... In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VCdimension of planar visibility systems is bounded by 23 if t ..."
Abstract - Cited by 13 (1 self) - Add to MetaCart
In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VCdimension of planar visibility systems is bounded by 23

VC-dimension of visibility on terrains

by James King - In Proc. 20th Canadian Conference on Comput. Geom , 2008
"... A guarding problem can naturally be modeled as a set system (U, S) in which the universe U of elements is the set of points we need to guard and our collection S of sets contains, for each potential guard g, the set of points from U seen by g. We prove bounds on the maximum VC-dimension of set syste ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
A guarding problem can naturally be modeled as a set system (U, S) in which the universe U of elements is the set of points we need to guard and our collection S of sets contains, for each potential guard g, the set of points from U seen by g. We prove bounds on the maximum VC-dimension of set

VC-Dimension of Sets of Permutations

by Ran Raz - Combinatorica , 2000
"... We define the VC-dimension of a set of permutations A ae S n to be the maximal k such that there exist distinct i 1 ; :::; i k 2 f1; :::; ng that appear in A in all possible linear orders, that is, every linear order of fi 1 ; :::; i k g is equivalent to the standard order of f(i 1 ); :::; (i k )g ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
We define the VC-dimension of a set of permutations A ae S n to be the maximal k such that there exist distinct i 1 ; :::; i k 2 f1; :::; ng that appear in A in all possible linear orders, that is, every linear order of fi 1 ; :::; i k g is equivalent to the standard order of f(i 1 ); :::; (i k

VC-Dimensions For Graphs

by Evangelos Kranakis, Danny Krizanc, Berthold Ruf, Jorge Urrutia, Gerhard J. Woeginger , 1994
"... We study set systems over the vertex set (or edge set) of some graph that are induced by special graph properties like clique, connectedness, path, star, tree, etc. We derive a variety of combinatorial and computational results on the VC (Vapnik-Chervonenkis) dimension of these set systems. For most ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
We study set systems over the vertex set (or edge set) of some graph that are induced by special graph properties like clique, connectedness, path, star, tree, etc. We derive a variety of combinatorial and computational results on the VC (Vapnik-Chervonenkis) dimension of these set systems

On the VC-Dimension of the Choquet Integral

by Eyke Hüllermeier, Ali Fallah Tehrani
"... Abstract. The idea of using the Choquet integral as an aggregation operator in machine learning has gained increasing attention in recent years, and a number of corresponding methods have already been proposed. Complementing these contributions from a more theoretical perspective, this paper address ..."
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addresses the following question: What is the VC dimension of the (discrete) Choquet integral when being used as a binary classifier? The VC dimension is a key notion in statistical learning theory and plays an important role in estimating the generalization performance of a learning method. Although we

On the complexity of approximating the vc dimension

by Elchanan Mossel - J. Comput. Syst. Sci , 2001
"... 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 show that this problem is • Σ p 3-hard to approximate to within a factor 2 − ɛ for any ɛ> 0, • approximable in AM to within a factor 2, and • AM-hard to a ..."
Abstract - Cited by 20 (3 self) - Add to MetaCart
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 show that this problem is • Σ p 3-hard to approximate to within a factor 2 − ɛ for any ɛ> 0, • approximable in AM to within a factor 2, and • AM
Results 1 - 10 of 541