One of the main open problems in the theory of multi-category margin classification is the form of the optimal dependency of a guaranteed risk on the number C of categories, the sample size m and the margin parameter γ. From a practical point of view, the theoretical analysis of generalization performance contributes to the development of new learning algorithms. In this paper, we focus only on the theoretical aspect of the question posed. More precisely, under minimal learnability assumptions, we derive a new risk bound for multi-category margin classifiers. We improve the dependency on C over the state of the art when the margin loss function considered satisfies the Lipschitz condition. We start with the basic supremum inequality that involves a Rademacher complexity as a capacity measure. This capacity measure is then linked to the metric entropy through the chaining method. In this context, our improvement is based on the introduction of a new combinatorial metric entropy bound.
BackgroundA well-known problem in cluster analysis is finding an optimal number of clusters reflecting the inherent structure of the data. PFClust is a partitioning-based clustering algorithm capable, unlike many widely-used clustering algorithms, of automatically proposing an optimal number of clusters for the data.ResultsThe results of tests on various types of data showed that PFClust can discover clusters of arbitrary shapes, sizes and densities. The previous implementation of the algorithm had already been successfully used to cluster large macromolecular structures and small druglike compounds. We have greatly improved the algorithm by a more efficient implementation, which enables PFClust to process large data sets acceptably fast.ConclusionsIn this paper we present a new optimized implementation of the PFClust algorithm that runs considerably faster than the original.
We control the probability of the uniform deviation between empirical and generalization performances of multi-category classifiers by an empirical L1-norm covering number when these performances are defined on the basis of the truncated hinge loss function. The only assumption made on the functions implemented by multi-category classifiers is that they are of bounded variation (BV ). For such classifiers, we derive the sample size estimate sufficient for the mentioned performances to be close with high probability. Particularly, we are interested in the dependency of this estimate on the number C of classes. To this end, first, we upper bound the scale-sensitive version of the VC-dimension, the fat-shattering dimension of sets of BV functions defined on R d which gives a O(ǫ −d ) as the scale ǫ goes to zero. Secondly, we provide a sharper decomposition result for the fat-shattering dimension in terms of C, which for sets of BV functions gives an improvement from O(C d 2 +1 ) to O(C ln 2 (C)). This improvement then propagates to the sample complexity estimate.
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