A Randomized Algorithm to Reduce the Support of Discrete Measures

• URL: http://arxiv.org/abs/2006.01757v2
• Date: Thu, 26 Nov 2020 09:12:31 GMT
• Title: A Randomized Algorithm to Reduce the Support of Discrete Measures
• Authors: Francesco Cosentino, Harald Oberhauser, Alessandro Abate
• Abstract summary: 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. 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 $Ngg n$ regime.
• Score: 79.55586575988292
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 \gg 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 $N\gg n$ regime. A Python implementation is available at \url{https://github.com/FraCose/Recombination_Random_Algos}.

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