Autoencoders have useful applications in high energy physics in both compression and anomaly detection, particularly for jets: collimated showers of particles produced in collisions such as those at the CERN Large Hadron Collider. We explore the use of graph-based autoencoders, which operate on jets in their "particle cloud" representations and can leverage the interdependencies among the particles within jets, for such tasks. Additionally, we develop a differentiable approximation to the energy mover's distance via a graph neural network, which may subsequently be used as a reconstruction loss function for autoencoders.