Quantification using permutation-invariant networks based on histograms

Quantification, also known as class prevalence estimation, is the supervised learning task in which a model is trained to predict the prevalence of each class in a given bag of examples. This paper investigates the application of deep neural networks for tasks of quantification in scenarios where it is possible to apply a symmetric supervised approach that eliminates the need for classification as an intermediate step, thus directly addressing the quantification problem. Additionally, it discusses existing permutation-invariant layers designed for set processing and assesses their suitability for quantification. Based on our analysis, we propose HistNetQ, a novel neural architecture that relies on a permutation-invariant representation based on histograms that is especially suited for quantification problems. Our experiments carried out in two standard competitions, which have become a reference in the quantification field, show that HistNetQ outperforms other deep neural network architectures designed for set processing, as well as the current state-of-the-art quantification methods. Furthermore, HistNetQ offers two significant advantages over traditional quantification methods: i) it does not require the labels of the training examples but only the prevalence values of a collection of training bags, making it applicable to new scenarios; and ii) it is able to optimize any custom quantification-oriented loss function.