# Gaussian Processes Sampling with Sparse Grids under Additive Schwarz Preconditioner

arXiv: https://arxiv.org/abs/2408.00206 · HTML: https://mlsanity.com/paper/2408.00206/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 97 references that are in the map.
- First posted on arXiv 1 Aug 2024.

## Builds on (most cited first, 30 of 30)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Exact Gaussian Processes on a Million Data Points](https://mlsanity.com/paper/1903.08114/) (2019)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [Preconditioning Kernel Matrices](https://mlsanity.com/paper/1602.06693/) (2016)
- [Bayesian Optimization with High-Dimensional Outputs](https://mlsanity.com/paper/2106.12997/) (2021)
- [Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent](https://mlsanity.com/paper/2306.11589/) (2023)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [Preconditioning for Scalable Gaussian Process Hyperparameter Optimization](https://mlsanity.com/paper/2107.00243/) (2021)
- [Kernel Interpolation for Scalable Online Gaussian Processes](https://mlsanity.com/paper/2103.01454/) (2021)
- [Scaled Vecchia approximation for fast computer-model emulation](https://mlsanity.com/paper/2005.00386/) (2020)
- [State Space Gaussian Processes with Non-Gaussian Likelihood](https://mlsanity.com/paper/1802.04846/) (2018)
- [Locally induced Gaussian processes for large-scale simulation experiments](https://mlsanity.com/paper/2008.12857/) (2020)
- [Parallelizable sparse inverse formulation Gaussian processes (SpInGP)](https://mlsanity.com/paper/1610.08035/) (2016)
- [Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat'ern Correlations](https://mlsanity.com/paper/2203.03116/) (2022)
- [Efficient Calibration for Imperfect Computer Models](https://mlsanity.com/paper/1507.07280/) (2015)
- [Fast prediction of deterministic functions using sparse grid experimental designs](https://mlsanity.com/paper/1402.6350/) (2014)
- [A Sparse Expansion For Deep Gaussian Processes](https://mlsanity.com/paper/2112.05888/) (2021)
- [Testing separability of space--time functional processes](https://mlsanity.com/paper/1509.07017/) (2015)
- [Bayesian Projected Calibration of Computer Models](https://mlsanity.com/paper/1803.01231/) (2018)
- [Gaussian Process Sampling and Optimization with Approximate Upper and Lower Bounds](https://mlsanity.com/paper/2110.12087/) (2021)
- [Kernel Interpolation with Sparse Grids](https://mlsanity.com/paper/2305.14451/) (2023)
- [Resolving Overlapping Convex Objects in Silhouette Images by Concavity Analysis and Gaussian Process](https://mlsanity.com/paper/1906.01049/) (2019)

## Abstract (arXiv)

Gaussian processes (GPs) are widely used in non-parametric Bayesian modeling, and play an important role in various statistical and machine learning applications. In a variety tasks of uncertainty quantification, generating random sample paths of GPs is of interest. As GP sampling requires generating high-dimensional Gaussian random vectors, it is computationally challenging if a direct method, such as the Cholesky decomposition, is used. In this paper, we propose a scalable algorithm for sampling random realizations of the prior and posterior of GP models. The proposed algorithm leverages inducing points approximation with sparse grids, as well as additive Schwarz preconditioners, which reduce computational complexity, and ensure fast convergence. We demonstrate the efficacy and accuracy of the proposed method through a series of experiments and comparisons with other recent works.