A Randomized Algorithm to Reduce the Support of Discrete Measures

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 on a subset of n + 1 of the original N atoms and has the same mean when integrated against each of the n functions. If N n this results in a huge reduction of complexity. We give a simple geometric characterization of barycenters via negative cones and derive a randomized algorithm that computes this 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 very beneficial in the N n regime. A Python implementation is available at https://github.com/FraCose/Recombination_Random_Algos .