| Abstract: |
| In recent years, optimal transport has witnessed a lot of applications in statistics, image processing, and machine learning. It provides a solid metric among histograms that incorporate the geometry (ground metric) of features. In this talk, we introduce the optimal transport on finite graphs, from which the probability simplex set forms a Riemannian manifold. We call it probability manifold. Various developments related to the optimal control problems in the probability manifold, e.g. discrete Fokker-Planck equation, Schrodinger equation/bridge problem and generalized Hopf-Lax formula will be sketched. Their connections with Shannon-Boltzmann entropy and Fisher information on graphs will be emphasized. Many applications will be discussed, including L1 Monge-Kantorovich problem, image segmentation, and population games. |
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