  We propose a framework for constructing and analyzing multiclass and
multioutput classification metrics, i.e., involving multiple, possibly
correlated multiclass labels. Our analysis reveals novel insights on the
geometry of feasible confusion tensors -- including necessary and sufficient
conditions for the equivalence between optimizing an arbitrary non-decomposable
metric and learning a weighted classifier. Further, we analyze averaging
methodologies commonly used to compute multioutput metrics and characterize the
corresponding Bayes optimal classifiers. We show that the plug-in estimator
based on this characterization is consistent and is easily implemented as a
post-processing rule. Empirical results on synthetic and benchmark datasets
support the theoretical findings.
