Sampling from Gaussian Processes

A Tutorial and Applications in Global Sensitivity Analysis and Optimization.

Gassian processes (GPs) naturally enable efficient sampling strategies that support informed decision-making under uncertainty.

We present the formulation and detailed implementation of two notable sampling methods -- random Fourier features and pathwise conditioning -- for generating posterior samples from GPs.

The code implements the random Fourier feature and pathwise conditioning methods.

Dependency Files

• ard_se_cov.m — Computing exact automatic relevance determination squared exponential covariance functions.
• chol2invchol.m — Computing the inverse of a symmetric positive definite matrix using its Cholesky decomposition.
• rff_ard_se_parameters.m — Sampling random points in the frequency and phase domains for random Fourier features (RFFs).
• rff_ard_se_posWeights_standard.m — Sampling posterior weights using Sherman-Morrison-Woodbury (SMW) formula and standard sampling of the weights.
• rff_ard_se_posWeights.m — Sampling posterior weights using Sherman-Morrison-Woodbury (SMW) formula and an eigendecomposition of SMW matrix. This is faster than rff_ard_se_posWeights_standard.m.
• rff_ard_se_features.m — Evaluating RFFs at query points of interest.
• rff_pc_se_paths.m — Drawing posterior, prior, and update paths using pathwise conditioning (PC) wih RFF prior.

Demo Files

• demo_RFF_covfnc.m — Computing approximate univariate SE covariance function for different number of RFFs.
• demo_RFF_samplepaths.m — Drawing several posterior samples using RFF method.
• demo_PC_decoupling_concepts.m — Drawing prior, update, and posterior samples using PC wih RFF prior.
• demo_PC_samplepaths.m — Drawing several posterior samples using PC wih RFF prior.

Requirements

• MATLAB R2018a or later (older versions might work, but are not tested).
• Required Toolbox: Statistics and Machine Learning Toolbox

Running the Experiments

1. Clone the repository:

   git clone https://github.com/UQUH/GPSampling.git
   cd GPSampling

2. Open MATLAB and set the repository folder as the working directory.

3. Run the demo script of interest.

Citation

If you found this tutorial helpful, please cite the following paper:

@misc{do2025sampling,
      title={Sampling from Gaussian Processes: A Tutorial and Applications in Global Sensitivity Analysis and Optimization}, 
      author={Do, Bach and Ajenifuja, Nafeezat A and Adebiyi, Taiwo A and Zhang, Ruda},
      year={2025},
      eprint={2507.14746},
      archivePrefix={arXiv},
      url={https://arxiv.org/abs/2507.14746}, 
}

The Team

The code is developed by the Uncertainty Quantification Lab at the University of Houston.

The contributors are:

• Bach Do
• Nafeezat Ajenifuja
• Taiwo Adebiyi
• Ruda Zhang