ERA ranking representability: The missing link between ordinal regression and multi-class classification

Can a multi-class classification model in some situations be simplified to an ordinal regression model without sacrificing performance? We try to answer this question from a theoretical point of view for one-versus-one multi-class ensembles. To that end, sufficient conditions are derived for which a one-versus-one ensemble becomes ranking representable, i.e. conditions for which the ensemble can be reduced to a ranking or ordinal regression model such that a similar performance on training data is measured. As performance measure, we use the area under the ROC curve (AUC) and its reformulation in terms of graphs. For the three-class case, this results in a new type of cycle transitivity for pairwise AUCs that can be verified by solving an integer quadratic program. Moreover, solving this integer quadratic program can be avoided, since its solution converges for an infinite data sample to a simple form, resulting in a deviation bound that becomes tighter with increasing sample size.