DDEQs: Distributional Deep Equilibrium Models through Wasserstein Gradient Flows

Deep Equilibrium Models (DEQs) are a class of implicit neural networks that solve for a fixed point of a neural network in their forward pass. Traditionally, DEQs take sequences as inputs, but have since been applied to a variety of data. In this work, we present Distributional Deep Equilibrium Models (DDEQs), extending DEQs to discrete measure inputs, such as sets or point clouds. We provide a theoretically grounded framework for DDEQs. Leveraging Wasserstein gradient flows, we show how the forward pass of the DEQ can be adapted to find fixed points of discrete measures under permutation-invariance, and derive adequate network architectures for DDEQs. In experiments, we show that they can compete with state-of-the-art models in tasks such as point cloud classification and point cloud completion, while being significantly more parameter-efficient.