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Given a discrete probability measure supported on $N$ atoms and a set of $n$real-valued functions, there exists a probability measure that is supported ona subset of $n+1$ of the original $N$ atoms and has the same mean whenintegrated against each of the $n$ functions. If $ N \gg n$ this results in ahuge reduction of complexity. We give a simple geometric characterization ofbarycenters via negative cones and derive a randomized algorithm that computesthis new measure by "greedy geometric sampling". We then study its properties,and benchmark it on synthetic and real-world data to show that it can be verybeneficial in the $N\gg n$ regime. A Python implementation is available at\url{https://github.com/FraCose/Recombination_Random_Algos}.

A Randomized Algorithm to Reduce the Support of Discrete Measures