Communications of the ACM
Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
A training algorithm for optimal margin classifiers
COLT '92 Proceedings of the fifth annual workshop on Computational learning theory
Efficient distribution-free learning of probabilistic concepts
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
Characterizations of learnability for classes of {0, …, n}-valued functions
Journal of Computer and System Sciences
A generalization of Sauer's lemma
Journal of Combinatorial Theory Series A
Machine Learning
Fat-shattering and the learnability of real-valued functions
Journal of Computer and System Sciences
Scale-sensitive dimensions, uniform convergence, and learnability
Journal of the ACM (JACM)
Generalization performance of support vector machines and other pattern classifiers
Advances in kernel methods
Support vector machines, reproducing kernel Hilbert spaces, and randomized GACV
Advances in kernel methods
A note on a scale-sensitive dimension of linear bounded functionals in Banach spaces
Theoretical Computer Science
Learning in Neural Networks: Theoretical Foundations
Learning in Neural Networks: Theoretical Foundations
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Choosing Multiple Parameters for Support Vector Machines
Machine Learning
On Learning Sets and Functions
Machine Learning
Entropy Numbers of Linear Function Classes
COLT '00 Proceedings of the Thirteenth Annual Conference on Computational Learning Theory
Reducing multiclass to binary: a unifying approach for margin classifiers
The Journal of Machine Learning Research
The Journal of Machine Learning Research
On the algorithmic implementation of multiclass kernel-based vector machines
The Journal of Machine Learning Research
In Defense of One-Vs-All Classification
The Journal of Machine Learning Research
Statistical Analysis of Some Multi-Category Large Margin Classification Methods
The Journal of Machine Learning Research
The Entire Regularization Path for the Support Vector Machine
The Journal of Machine Learning Research
Estimation of Dependences Based on Empirical Data: Springer Series in Statistics (Springer Series in Statistics)
On the Consistency of Multiclass Classification Methods
The Journal of Machine Learning Research
Fast rates for support vector machines
COLT'05 Proceedings of the 18th annual conference on Learning Theory
IEEE Transactions on Information Theory
Structural risk minimization over data-dependent hierarchies
IEEE Transactions on Information Theory
Model Selection: Beyond the Bayesian/Frequentist Divide
The Journal of Machine Learning Research
Estimating the class posterior probabilities in protein secondary structure prediction
PRIB'11 Proceedings of the 6th IAPR international conference on Pattern recognition in bioinformatics
Analysis of a multi-category classifier
Discrete Applied Mathematics
A generic model of multi-class support vector machine
International Journal of Intelligent Information and Database Systems
A note on extending generalization bounds for binary large-margin classifiers to multiple classes
ECML PKDD'12 Proceedings of the 2012 European conference on Machine Learning and Knowledge Discovery in Databases - Volume Part I
Cascading discriminant and generative models for protein secondary structure prediction
PRIB'12 Proceedings of the 7th IAPR international conference on Pattern Recognition in Bioinformatics
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In the context of discriminant analysis, Vapnik's statistical learning theory has mainly been developed in three directions: the computation of dichotomies with binary-valued functions, the computation of dichotomies with real-valued functions, and the computation of polytomies with functions taking their values in finite sets, typically the set of categories itself. The case of classes of vector-valued functions used to compute polytomies has seldom been considered independently, which is unsatisfactory, for three main reasons. First, this case encompasses the other ones. Second, it cannot be treated appropriately through a naïve extension of the results devoted to the computation of dichotomies. Third, most of the classification problems met in practice involve multiple categories. In this paper, a VC theory of large margin multi-category classifiers is introduced. Central in this theory are generalized VC dimensions called the γ-Ψ-dimensions. First, a uniform convergence bound on the risk of the classifiers of interest is derived. The capacity measure involved in this bound is a covering number. This covering number can be upper bounded in terms of the γ-Ψ-dimensions thanks to generalizations of Sauer's lemma, as is illustrated in the specific case of the scale-sensitive Natarajan dimension. A bound on this latter dimension is then computed for the class of functions on which multi-class SVMs are based. This makes it possible to apply the structural risk minimization inductive principle to those machines.