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 [InterStat 5 (2004) 1-6] 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.

Best finite constrained approximations of one-dimensional probabilities

“…Among the best approximations of µ, given p ∈ Π n , the case of uniform approximations, i.e., p = u n , arguably is the most important. In this case, Theorem 5.9 has the following corollary; see also [25,Thm.2].…”

“…The special case of best uniform approximations is of considerable interest in itself: In statistics, when dealing with empirical data sets, practical considerations may demand that all atoms have equal weights, or at least that they be integer multiples of one fixed unit weight [2]. Also, best uniform approximations are close analogues of support points [25], the latter being minimizers relative to a slightly different metric (energy distance). One may thus view δ un • as a quasi Monte Carlo (MC) tool that minimizes the integration error bound | f dµ − n j=1 f (x j )| ≤ Lip (f ) d 1 (δ un • , µ) for a wide class of functions (cf.…”

“…In computational mathematics, best uniform approximations also arise naturally in the form of restricted MC methods, and a basic question is how performance of the latter compares to general (non-restricted) MC methods, that is, how d r (δ un • , µ) compares to d r (δ •,n • , µ) as n → ∞; see, e.g., [14,15] and the many references therein. Restricted MC methods have recently found applications in "big-data" problems in Bayesian statistics [25] and the numerical solution of SDE, notably in mathematical finance [14,16,28].…”

Quadrature Points via Heat Kernel Repulsion

“…We have shown the method to be competitive against methods that have been explicitly tailored for the sphere and the torus. We have not compared the method to methods on other manifolds since there are not that many universal design rules (though this is becoming a more popular topic of research, see [10,35]). Our method is superior to points minimizing the Riesz energy; this is, considering the singularities of the Fourier transform of the Riesz kernel, perhaps not surprising.…”

Interleaved lattice-based maximin distance designs

“…Consequently, points ofD(L, s) represent [0, 1] p better than that of D(L, s) generated via (6) in not exaggerating the near-boundary regions. Thus, they can be seen as support points and may be useful in some applications (Mak and Joseph, 2018). On the other hand, D(L, s) has higher separation distance and may be more suitable to the emulation problem for which denser points in the near-boundary regions is desired (Dette and Pepelyshev, 2010).…”