  In imbalanced multi-class classification problems, the misclassification rate
as an error measure may not be a relevant choice. Several methods have been
developed where the performance measure retained richer information than the
mere misclassification rate: misclassification costs, ROC-based information,
etc. Following this idea of dealing with alternate measures of performance, we
propose to address imbalanced classification problems by using a new measure to
be optimized: the norm of the confusion matrix. Indeed, recent results show
that using the norm of the confusion matrix as an error measure can be quite
interesting due to the fine-grain informations contained in the matrix,
especially in the case of imbalanced classes. Our first contribution then
consists in showing that optimizing criterion based on the confusion matrix
gives rise to a common background for cost-sensitive methods aimed at dealing
with imbalanced classes learning problems. As our second contribution, we
propose an extension of a recent multi-class boosting method --- namely
AdaBoost.MM --- to the imbalanced class problem, by greedily minimizing the
empirical norm of the confusion matrix. A theoretical analysis of the
properties of the proposed method is presented, while experimental results
illustrate the behavior of the algorithm and show the relevancy of the approach
compared to other methods.
