# Quantification using Permutation-Invariant Networks based on Histograms

arXiv: https://arxiv.org/abs/2403.15123 · HTML: https://mlsanity.com/paper/2403.15123/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 51 references that are in the map.
- First posted on arXiv 22 Mar 2024.

## Builds on (most cited first, 18 of 18)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Towards a Neural Statistician](https://mlsanity.com/paper/1606.02185/) (2016)
- [Janossy Pooling: Learning Deep Permutation-Invariant Functions for Variable-Size Inputs](https://mlsanity.com/paper/1811.01900/) (2018)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [Histogram Layers for Texture Analysis](https://mlsanity.com/paper/2001.00215/) (2020)
- [Quantification under prior probability shift: the ratio estimator and its extensions](https://mlsanity.com/paper/1807.03929/) (2018)
- [QuaPy: A Python-Based Framework for Quantification](https://mlsanity.com/paper/2106.11057/) (2021)
- [Evaluation Measures for Quantification: An Axiomatic Approach](https://mlsanity.com/paper/1809.01991/) (2018)
- [Class Prior Estimation under Covariate Shift: No Problem?](https://mlsanity.com/paper/2206.02449/) (2022)
- [Learnable Histogram: Statistical Context Features for Deep Neural Networks](https://mlsanity.com/paper/1804.09398/) (2018)
- [DeepHist: Differentiable Joint and Color Histogram Layers for Image-to-Image Translation](https://mlsanity.com/paper/2005.03995/) (2020)
- [A Recurrent Neural Network for Sentiment Quantification](https://mlsanity.com/paper/1809.00836/) (2018)
- [Tweet Sentiment Quantification: An Experimental Re-Evaluation](https://mlsanity.com/paper/2011.08091/) (2020)
- [Quantification in-the-wild: data-sets and baselines](https://mlsanity.com/paper/1510.04811/) (2015)
- [Differentiable Histogram with Hard-Binning](https://mlsanity.com/paper/2012.06311/) (2020)

## Built on by (most cited first, 3 of 3)

- [On the Interconnections of Calibration, Quantification, and Classifier Accuracy Prediction under Dataset Shift](https://mlsanity.com/paper/2505.11380/) (2025)
- [Optimal Transport-based Permutation-Invariant Bayesian Optimization of Offshore Wind Farm Layouts](https://mlsanity.com/paper/2606.00009/) (2026)
- [Class-Conditioned Gaussian Mixture Modeling for Imbalanced Time Series Quantification](https://mlsanity.com/paper/2608.21473/) (2026)

## Abstract (arXiv)

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 to tasks of quantification in scenarios where it is possible to apply a symmetric supervised approach that eliminates the need for classification as an intermediary step, directly addressing the quantification problem. Additionally, it discusses existing permutation-invariant layers designed for set processing and assesses their suitability for quantification. In light of our analysis, we propose HistNetQ, a novel neural architecture that relies on a permutation-invariant representation based on histograms that is specially suited for quantification problems. Our experiments carried out in the only quantification competition held to date, show that HistNetQ outperforms other deep neural architectures devised for set processing, as well as the 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.