# Support points

@article{Mak2018SupportP, title={Support points}, author={Simon Mak and V. Roshan Joseph}, journal={The Annals of Statistics}, year={2018} }

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…

## Figures and Tables from this paper

## 65 Citations

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

- Computer ScienceSIAM/ASA J. Uncertain. Quantification
- 2020

Connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures are investigated.

Periodic version of the minimax distance criterion for Monte Carlo integration

- MathematicsAdv. Eng. Softw.
- 2020

Best finite constrained approximations of one-dimensional probabilities

- MathematicsJ. Approx. Theory
- 2019

Stein Points

- Computer ScienceICML
- 2018

The empirical results demonstrate that Stein Points enable accurate approximation of the posterior at modest computational cost, and theoretical results are provided to establish convergence of the method.

Generating Point Configurations via Hypersingular Riesz Energy with an External Field

- MathematicsSIAM J. Math. Anal.
- 2017

A method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of \mathbb{R}^{p} $ and the first-order asymptotic values of minimal energy are obtained.

Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

- MathematicsJ. Approx. Theory
- 2022

Stein Point Markov Chain Monte Carlo

- Computer ScienceICML
- 2019

This paper removes the need to solve this optimisation problem by selecting each new point based on a Markov chain sample path, which significantly reduces the computational cost of Stein Points and leads to a suite of algorithms that are straightforward to implement.

An Efficient and Distribution-Free Two-Sample Test Based on Energy Statistics and Random Projections

- Computer Science, Mathematics
- 2017

This paper develops an efficient algorithm with complexity $O(N \log N)$ for computing energy statistics in univariate cases and introduces a two-sample test based on energy statistics and random projections, which enjoys the computational complexity.

Population Quasi-Monte Carlo

- Computer ScienceJournal of Computational and Graphical Statistics
- 2022

A new Population Quasi-Monte Carlo (PQMC) framework is proposed, which integrates quasi-monte Carlo ideas within the sampling and adaptation steps of PMC, and an efficient covariance adaptation strategy for multivariate normal proposals is developed.

Performance analysis of greedy algorithms for minimising a Maximum Mean Discrepancy

- Computer ScienceArXiv
- 2021

It is shown that the ﬁnite-sample-size approximation error, measured by the MMD, decreases as 1 /n for SBQ and also for kernel herding and greedy MMD minimisation when using a suitable step-size sequence.

## References

SHOWING 1-10 OF 89 REFERENCES

High-dimensional integration: The quasi-Monte Carlo way*†

- Computer ScienceActa Numerica
- 2013

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s…

Minimax designs using clustering

- Computer Science, Mathematics
- 2016

A new clustering-based method is presented for computing minimax designs on any convex and bounded design region and it is shown that the proposed algorithm provides improved minimax performance over existing methods on a variety of design regions.

Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration

- Computer Science
- 2010

This comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research.

ASYMPTOTICALLY OPTIMAL REPRESENTATIVE POINTS OF BIVARIATE RANDOM VECTORS

- Mathematics
- 2000

Optimal representative points (also called principal points) of scalar ran- dom variables are known in several cases and asymptotically optimal representa- tive points are known for all densities, as…

Convex analysis approach to d.c. programming: Theory, Algorithm and Applications

- Computer Science
- 1997

A thorough study on convex analysis approach to d.C.c. (difierence of convex functions) programming and gives the State of the Art results and the application of the DCA to solving a lot of important real-life d.c., polyhedral programming problems.

Fast CBC construction of randomly shifted lattice rules achieving O(n-1+δ) convergence for unbounded integrands over R5 in weighted spaces with POD weights

- MathematicsJ. Complex.
- 2014

Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2002

The shifts in shifted rank-1 lattice rules, a special class of quasi-Monte Carlo methods, proposed for the integration of functions belonging to certain "weighted" Sobolev spaces, are generated randomly to allow probabilistic estimates for the error in a given integral.