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Abstract:
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We discuss recent developments in statistical data analysis based on empirical optimal transport (EOT). The EOT plan finds the most efficient way to match data sets, e.g. w.r.t. physical energy, spatial distance or economic cost. To make EOT and variants thereof useful for daily data analysis, two bottlenecks have to be overcome: A computational burden as well as a lack of statistical understanding and methodology. Both issues will be addressed in this talk. In the first part we will discuss distributional limit laws and probabilistic bounds for EOT plans and distances on discrete spaces and its use for inference and simulation. In the second part we investigate resampling as a computational scheme. We provide bounds which allow to balance computational speed and statistical accuracy. Finally, we discuss extensions such as OT based barycenters and data analysis in the Gromov-Wasserstein space. The presented methodology is illustrated by computer exeriments and in different examples, mainly from super-resolution cell microscopy.
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