Learning from Networks of Distributions

There are multiple settings where it is more natural to represent data not as individual training points (vectors), but as probability distributions. In some cases, we may be provided with samples from the distribution (e.g., ecological inference, multi-instance learning) and in other cases, we may have access to parametric or semi-parametric specifications of a distribution. These latter specifications are often derived from a predictive algorithm. Our goal is to learn from these (empirical) distributions, so that we can perform a task such as regression or classification - mapping an input distribution to a real-valued output or a label. In this paper, we consider the case where, in addition to the observed distributions, we have access to relational data in the form of a network or graph. The graph specifies structural relationships between the observed distributions. Our goal is to design a learning algorithm that takes into account the graph structure as well as the information from the distributions. We describe a graph neural network architecture that performs information aggregation to learn. We demonstrate how our algorithm can be used in an example application.