Support points

TL;DR

Support points are a novel method for representing continuous distributions with a finite set of points by minimizing the energy distance, leading to better approximation and integration performance compared to traditional methods.

Contribution

The paper introduces support points, a new approach to compact continuous distributions using energy distance minimization, with theoretical convergence guarantees and efficient algorithms.

Findings

Support points converge in distribution to the target distribution.

Support points improve integration accuracy over Monte Carlo methods.

Support points enhance uncertainty quantification and MCMC sample compression.

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 Sz\'ekely and Rizzo (2004) for testing goodness-of-fit. The energy distance has two appealing features. First, its distance-based 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 difference-of-convex 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 Quasi-Monte 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.