We consider the statistical problem of recovering the optimal transport matching between two discrete measures from noisy convolutions of these measures with a smooth kernel. This question is motivated by super-resolution microscopy applications, where optimal transport matchings have recently been proposed as a figure of merit for quantifying the spatial proximity between discrete signals under observation. Our main result characterizes the minimax risk of estimating a discrete measure under the Wasserstein distance in this observation model. As a byproduct, we derive the minimax rate of estimating optimal transport matchings in this setting. These rates are achieved by simple estimators based on maximum likelihood estimation.
This talk is based on joint work with Shayan Hundrieser, Danila Litskevich, and Axel Munk.
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