Support points
Abstract
This paper introduces a new way to compact a continuous probability distribution $F$ into a set of representative points called support points. These points are obtained by minimizing the energy distance, a statistical potential measure initially proposed by Székely and Rizzo (2004) for testing goodnessoffit. The energy distance has two appealing features. First, its distancebased structure allows us to exploit the duality between powers of the Euclidean distance and its Fourier transform for theoretical analysis. Using this duality, we show that support points converge in distribution to $F$, and enjoy an improved error rate to Monte Carlo for integrating a large class of functions. Second, the minimization of the energy distance can be formulated as a differenceofconvex program, which we manipulate using two algorithms to efficiently generate representative point sets. In simulation studies, support points provide improved integration performance to both Monte Carlo and a specific QuasiMonte Carlo method. Two important applications of support points are then highlighted: (a) as a way to quantify the propagation of uncertainty in expensive simulations, and (b) as a method to optimally compact Markov chain Monte Carlo (MCMC) samples in Bayesian computation.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.01811
 Bibcode:
 2016arXiv160901811M
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Methodology;
 62E17
 EPrint:
 Accepted, Annals of Statistics