A major trajectory in the development of statistical learning has been the expansion of mathematical spaces underlying observed data, extending from numbers to vectors, functions, and beyond. This expansion has fostered significant theoretical and computational breakthroughs. One notable direction involves analyzing sets where each set becomes an object of interest for inference. This perspective accommodates the intrinsic and non-ignorable heterogeneity inherent in data-generating processes. Among various theoretical frameworks to analyze sets, a principled approach is viewing a set as an empirical measure. In this talk, I revisit the concept of the median - a robust alternative to the mean as a centroid - and introduce a novel extension of this concept within the space of probability measures under the framework of optimal transport. I will present theoretical results and a generic computational pipeline that leverages existing algorithmic developments in the field, with examples. Furthermore, the potential benefits of this novel approach for scalable inference and scientific discovery will be explored.
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