Probability distributions represent natural ways to encode knowledge about the frequency of occurrence of events. Optimal transport (OT) provides a natural way to compare and interpolate between different probability distributions. For example, these ideas can be used to compare two images in a face recognition algorithm or to interpolate between two shapes in a computer graphics application. Typical approaches to solving discrete optimal transport problems rely on either solving an assignment problem or slightly changing the objective to make the problem more tractable. These approaches are either computationally expensive, not adaptable to specific types of data, or only provide rough approximations of the true solution. We investigate ways to adapt approaches inspired by fluid dynamics, differential equations, and geometry to discrete domains (for example, graphs). Specifically, we are looking at computationally inexpensive methods for optimal transport on graphs with various applications to modelling and analyzing the evolution of probability distributions on discrete domains.
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