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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.<\/jats:p>","DOI":"10.1007\/s00521-024-10721-1","type":"journal-article","created":{"date-parts":[[2024,12,14]],"date-time":"2024-12-14T08:25:16Z","timestamp":1734164716000},"page":"3505-3520","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quantification using permutation-invariant networks based on histograms"],"prefix":"10.1007","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4527-6698","authenticated-orcid":false,"given":"Olaya","family":"P\u00e9rez-Mon","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alejandro","family":"Moreo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Juan Jos\u00e9 del","family":"Coz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pablo","family":"Gonz\u00e1lez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,12,14]]},"reference":[{"key":"10721_CR1","unstructured":"Beijbom O, Hoffman J, Yao E, Darrell T, Rodriguez-Ramirez A, Gonzalez-Rivero M, Guldberg OH- (2015) Quantification in-the-wild: data-sets and baselines. arXiv:1510.04811 [cs] (2015). arXiv: 1510.04811"},{"key":"10721_CR2","doi-asserted-by":"publisher","unstructured":"Forman G (2006) Quantifying trends accurately despite classifier error and class imbalance. 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