# The Earth Mover's Pinball Loss: Quantiles for Histogram-Valued Regression

arXiv: https://arxiv.org/abs/2106.02051 · HTML: https://mlsanity.com/paper/2106.02051/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 65 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Ray: A Distributed Framework for Emerging AI Applications](https://mlsanity.com/paper/1712.05889/) (2017)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Label Distribution Learning](https://mlsanity.com/paper/1408.6027/) (2014)
- [PCANet: A Simple Deep Learning Baseline for Image Classification?](https://mlsanity.com/paper/1404.3606/) (2014)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [Learning Deep Embeddings with Histogram Loss](https://mlsanity.com/paper/1611.00822/) (2016)
- [Squared Earth Mover's Distance-based Loss for Training Deep Neural Networks](https://mlsanity.com/paper/1611.05916/) (2016)
- [Histogram Layers for Texture Analysis](https://mlsanity.com/paper/2001.00215/) (2020)
- [Cosmological N-body simulations: a challenge for scalable generative models](https://mlsanity.com/paper/1908.05519/) (2019)
- [Predicting Aesthetic Score Distribution through Cumulative Jensen-Shannon Divergence](https://mlsanity.com/paper/1708.07089/) (2017)
- [Learnable Histogram: Statistical Context Features for Deep Neural Networks](https://mlsanity.com/paper/1804.09398/) (2018)
- [A Novel Smoothed Loss and Penalty Function for Noncrossing Composite Quantile Estimation via Deep Neural Networks](https://mlsanity.com/paper/1909.12122/) (2019)
- [HistoNet: Predicting size histograms of object instances](https://mlsanity.com/paper/1912.05227/) (2019)
- [Model-aware Quantile Regression for Discrete Data](https://mlsanity.com/paper/1804.03714/) (2018)
- [Continuous Histogram Loss: Beyond Neural Similarity](https://mlsanity.com/paper/2004.02830/) (2020)
- [Semi-parametric γ-ray modeling with Gaussian processes and variational inference](https://mlsanity.com/paper/2010.10450/) (2020)

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

- [A deep learning framework for jointly extracting spectra and source-count distributions in astronomy](https://mlsanity.com/paper/2401.03336/) (2024)
- [Locally orderless networks](https://mlsanity.com/paper/2406.13514/) (2024)
- [Bridging the Compression-Precision Paradox: A Hybrid Architecture for Clinical EEG Report Generation with Guaranteed Measurement Accuracy](https://mlsanity.com/paper/2602.10544/) (2026)

## Abstract (arXiv)

Although ubiquitous in the sciences, histogram data have not received much attention by the Deep Learning community. Whilst regression and classification tasks for scalar and vector data are routinely solved by neural networks, a principled approach for estimating histogram labels as a function of an input vector or image is lacking in the literature. We present a dedicated method for Deep Learning-based histogram regression, which incorporates cross-bin information and yields distributions over possible histograms, expressed by τ-quantiles of the cumulative histogram in each bin. The crux of our approach is a new loss function obtained by applying the pinball loss to the cumulative histogram, which for 1D histograms reduces to the Earth Mover's distance (EMD) in the special case of the median (τ = 0.5), and generalizes it to arbitrary quantiles. We validate our method with an illustrative toy example, a football-related task, and an astrophysical computer vision problem. We show that with our loss function, the accuracy of the predicted median histograms is very similar to the standard EMD case (and higher than for per-bin loss functions such as cross-entropy), while the predictions become much more informative at almost no additional computational cost.