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 $\tau$-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 ($\tau = 0.5$), 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.

本文提出一种基于深度学习的直方图回归方法，它包含交叉区间信息并产生可能直方图的分布，并将针球损失应用于累积直方图中以获得新的损失函数，该函数在预测中位数直方图的准确度上类似于标准的 Earth Mover's distance(EMD)。（$\tau=0.5$），并将其推广到任意分位数。

移动地球销球器损失: 直方图值回归的分位数