# Support points

arXiv: https://arxiv.org/abs/1609.01811 · HTML: https://mlsanity.com/paper/1609.01811/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2016.

## Built on by (most cited first, 30 of 39)

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- [Performance analysis of greedy algorithms for minimising a Maximum Mean Discrepancy](https://mlsanity.com/paper/2101.07564/) (2021)
- [Risk-Aware Control and Optimization for High-Renewable Power Grids](https://mlsanity.com/paper/2204.00950/) (2022)
- [Kernel Two-Sample Tests in High Dimension: Interplay Between Moment Discrepancy and Dimension-and-Sample Orders](https://mlsanity.com/paper/2201.00073/) (2021)
- [Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation](https://mlsanity.com/paper/2203.03185/) (2022)
- [An optimal transport approach for selecting a representative subsample with application in efficient kernel density estimation](https://mlsanity.com/paper/2206.01182/) (2022)
- [Compressed Monte Carlo with application in particle filtering](https://mlsanity.com/paper/2107.08459/) (2021)
- [A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel](https://mlsanity.com/paper/2111.10722/) (2021)
- [Model-free Subsampling Method Based on Uniform Designs](https://mlsanity.com/paper/2209.03617/) (2022)
- [A Global-Local Approximation Framework for Large-Scale Gaussian Process Modeling](https://mlsanity.com/paper/2305.10158/) (2023)
- [Enhancing Semi-Supervised Learning via Representative and Diverse Sample Selection](https://mlsanity.com/paper/2409.11653/) (2024)
- [Towards Representation Learning for Weighting Problems in Design-Based Causal Inference](https://mlsanity.com/paper/2409.16407/) (2024)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Reproducible, incremental representation learning with Rosetta VAE](https://mlsanity.com/paper/2201.05206/) (2022)
- [Bayes Hilbert Spaces for Posterior Approximation](https://mlsanity.com/paper/2304.09053/) (2023)
- [Monte Carlo with kernel-based Gibbs measures: Guarantees for probabilistic herding](https://mlsanity.com/paper/2402.11736/) (2024)
- [Using Low-Discrepancy Points for Data Compression in Machine Learning: An Experimental Comparison](https://mlsanity.com/paper/2407.07450/) (2024)
- [Neural Low-Discrepancy Sequences](https://mlsanity.com/paper/2510.03745/) (2025)
- [Thinned Mean Field Langevin Dynamics](https://mlsanity.com/paper/2605.28589/) (2026)
- [Spatio-Temporal Gaussian Process for Building Terrain-Incorporating Wind Power Curves](https://mlsanity.com/paper/2607.00051/) (2026)
- [Sample selection from a given dataset to validate machine learning models](https://mlsanity.com/paper/2104.14401/) (2021)
- [Optimal Sub-sampling to Boost Power of Kernel Sequential Change-point Detection](https://mlsanity.com/paper/2210.15060/) (2022)
- [A model-free subdata selection method for classification](https://mlsanity.com/paper/2404.19127/) (2024)
- [Double Machine Learning for Adaptive Causal Representation in High-Dimensional Data](https://mlsanity.com/paper/2411.14665/) (2024)
- [Towards Clinical Practice in CT-Based Pulmonary Disease Screening: An Efficient and Reliable Framework](https://mlsanity.com/paper/2412.01525/) (2024)
- [Modeling High-Resolution Spatio-Temporal Wind with Deep Echo State Networks and Stochastic Partial Differential Equations](https://mlsanity.com/paper/2412.07265/) (2024)

## Abstract (arXiv)

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.