# Positively Weighted Kernel Quadrature via Subsampling

arXiv: https://arxiv.org/abs/2107.09597 · HTML: https://mlsanity.com/paper/2107.09597/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 84 references that are in the map.
- First posted on arXiv 20 Jul 2021.

## Builds on (most cited first, 16 of 16)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Kernel interpolation with continuous volume sampling](https://mlsanity.com/paper/2002.09677/) (2020)
- [Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees](https://mlsanity.com/paper/1506.02681/) (2015)
- [Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings](https://mlsanity.com/paper/1709.00147/) (2017)
- [Convergence guarantees for kernel-based quadrature rules in misspecified settings](https://mlsanity.com/paper/1605.07254/) (2016)
- [Black-box Importance Sampling](https://mlsanity.com/paper/1610.05247/) (2016)
- [A Bayes-Sard Cubature Method](https://mlsanity.com/paper/1804.03016/) (2018)
- [Semi-Exact Control Functionals From Sard's Method](https://mlsanity.com/paper/2002.00033/) (2020)
- [A Randomized Algorithm to Reduce the Support of Discrete Measures](https://mlsanity.com/paper/2006.01757/) (2020)
- [A Statistical Perspective on Coreset Density Estimation](https://mlsanity.com/paper/2011.04907/) (2020)
- [Methods for Sparse and Low-Rank Recovery under Simplex Constraints](https://mlsanity.com/paper/1605.00507/) (2016)

## Built on by (most cited first, 25 of 25)

- [Looping in the Human Collaborative and Explainable Bayesian Optimization](https://mlsanity.com/paper/2310.17273/) (2023)
- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [Sampling-based Nystr"om Approximation and Kernel Quadrature](https://mlsanity.com/paper/2301.09517/) (2023)
- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Kernel quadrature with randomly pivoted Cholesky](https://mlsanity.com/paper/2306.03955/) (2023)
- [Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling](https://mlsanity.com/paper/2311.13548/) (2023)
- [Bayesian Model Selection of Lithium-Ion Battery Models via Bayesian Quadrature](https://mlsanity.com/paper/2210.17299/) (2022)
- [SOBER: Highly Parallel Bayesian Optimization and Bayesian Quadrature over Discrete and Mixed Spaces](https://mlsanity.com/paper/2301.11832/) (2023)
- [Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approach](https://mlsanity.com/paper/2306.05843/) (2023)
- [WildCat: Near-Linear Attention in Theory and Practice](https://mlsanity.com/paper/2602.10056/) (2026)
- [Nested Expectations with Kernel Quadrature](https://mlsanity.com/paper/2502.18284/) (2025)
- [When can Regression-Adjusted Control Variates Help? Rare Events, Sobolev Embedding and Minimax Optimality](https://mlsanity.com/paper/2305.16527/) (2023)
- [A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Lifting](https://mlsanity.com/paper/2404.12219/) (2024)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Theoretical Refinement of CLIP by Utilizing Linear Structure of Optimal Similarity](https://mlsanity.com/paper/2510.15508/) (2025)
- [To discretize continually: Mean shift interacting particle systems for Bayesian inference](https://mlsanity.com/paper/2605.14142/) (2026)
- [Bayesian Quadrature for Neural Ensemble Search](https://mlsanity.com/paper/2303.08874/) (2023)
- [Compressed online Sinkhorn](https://mlsanity.com/paper/2310.05019/) (2023)
- [Policy Gradient with Kernel Quadrature](https://mlsanity.com/paper/2310.14768/) (2023)
- [Solving the Cold Start Problem on One's Own as an End User via Preference Transfer](https://mlsanity.com/paper/2502.12398/) (2025)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Fast Empirical Scenarios](https://mlsanity.com/paper/2307.03927/) (2023)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)
- [Amortized quadrature for posterior expectations in inverse problems](https://mlsanity.com/paper/2606.15871/) (2026)

## Abstract (arXiv)

We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.