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Title: Parameter Selection in Fermat Distances: Navigating Geometry and Noise

Abstract: Fermat distances are metrics that can be inferred from datasets. In their empirical (microscopic) version, they are defined following the model of first passage Euclidean percolation. The macroscopic (population) version establishes a metric that depends on the density from which the points were sampled. This characteristic makes these distances useful for various tasks such as classification, clustering, topological learning, optimal transport, and Wasserstein barycenter computation,...

The distances hinge on a parameter that requires careful selection. Throughout this presentation, we will delve into how these distances can effectively determine clusters at both the population and empirical levels, supported by consistency theorems. Moreover, we will leverage this understanding to gain insights on the choice of the parameter. The exploration of the asymptotic behavior of these distances translates into first-pass percolation problems, many of which remain unsolved.