UC San Diego

Many modern problems in data science involve data that is fundamentally measure-valued. In these situations, natural questions arise about how one should perform standard tasks from machine learning and numerical analysis. The tools of optimal transport point to intuitive analogs of these tasks by viewing the measures themselves as single points on the Wasserstein manifold. Then, by computing optimal transport maps from a fixed reference, we can embed each measure into a common linear space (the tangent space of the Wasserstein manifold at the fixed reference), which allows us to apply out-of-the-box methods on the embeddings. When the data is static, these methods can be used to build classifiers, generate new measures from particular classes, efficiently compute clusters, and produce visualizations via dimension reduction. In this dissertation, we extend this technique of ``linearized optimal transport'' to the case where our measure-valued data is evolving in time. In this case, we interpret our data as a curve on the Wasserstein manifold, and we use optimal transport to compute tangent vectors to this curve. These tangent vectors allow us to produce geodesics in specified directions, which can be used both for interpolation between measures and for extrapolation to predict future evolution. We rigorously develop this extrapolation as an analog of Euler's method in Euclidean space, and we use example particle simulators to demonstrate how this approach can accurately approximate (sufficiently nice) curves of evolving measures. We further develop the analog of Newton's method for finding steady states of particle models, connect it to our Euler timestepper, and demonstrate how steady-state distributions can be efficiently computed using Newton--Krylov methods. As a final application, we observe that textual prompt embeddings in the Stable Diffusion pipeline can be treated as discrete uniform measures, and we show that using Wasserstein geodesics to interpolate between two prompts gives a more ``natural'' way to interpolate between output images.