Scale-sensitive dimensions and skeleton estimates for classification (Q1265744)

The classical binary classification problem is investigated when it is known in advance that the posterior probability function (or regression function) belongs to some class of functions. The authors introduce and analyze methods which effectively exploit this knowledge. These methods are based on minimizing the empirical risk over a carefully selected ``skeleton'' of the class of regression functions. The skeletons are coverings of the class based on metrics, especially fitted for classification. A new scale-sensitive dimension is introduced which is more suitable for the studied classification problem than other, previously defined, dimension measures. This fact is demonstrated by performance bounds for the skeleton estimates in terms of the new dimension.