  Quantification is the machine learning task of estimating test-data class
proportions that are not necessarily similar to those in training. Apart from
its intrinsic value as an aggregate statistic, quantification output can also
be used to optimize classifier probabilities, thereby increasing classification
accuracy. We unify major quantification approaches under a constrained
multi-variate regression framework, and use mathematical programming to
estimate class proportions for different loss functions. With this modeling
approach, we extend existing binary-only quantification approaches to
multi-class settings as well. We empirically verify our unified framework by
experimenting with several multi-class datasets including the Stanford
Sentiment Treebank and CIFAR-10.
