  The Earth mover's distance (EMD) is a useful metric for image recognition and
classification, but its usual implementations are not differentiable or too
slow to be used as a loss function for training other algorithms via gradient
descent. In this paper, we train a convolutional neural network (CNN) to learn
a differentiable, fast approximation of the EMD and demonstrate that it can be
used as a substitute for computing-intensive EMD implementations. We apply this
differentiable approximation in the training of an autoencoder-inspired neural
network (encoder NN) for data compression at the high-luminosity LHC at CERN.
The goal of this encoder NN is to compress the data while preserving the
information related to the distribution of energy deposits in particle
detectors. We demonstrate that the performance of our encoder NN trained using
the differentiable EMD CNN surpasses that of training with loss functions based
on mean squared error.
