# Permutation and Grouping Methods for Sharpening Gaussian Process Approximations

arXiv: https://arxiv.org/abs/1609.05372 · HTML: https://mlsanity.com/paper/1609.05372/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2016.

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

- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Scalable Bayesian Optimization Using Vecchia Approximations of Gaussian Processes](https://mlsanity.com/paper/2203.01459/) (2022)
- [Variational Nearest Neighbor Gaussian Process](https://mlsanity.com/paper/2202.01694/) (2022)
- [Scalable Gaussian-process regression and variable selection using Vecchia approximations](https://mlsanity.com/paper/2202.12981/) (2022)
- [Iterative Methods for Vecchia-Laplace Approximations for Latent Gaussian Process Models](https://mlsanity.com/paper/2310.12000/) (2023)
- [Sensitivity Prewarping for Local Surrogate Modeling](https://mlsanity.com/paper/2101.06296/) (2021)
- [Deep Generative Spatiotemporal Engression for Probabilistic Forecasting of Epidemics](https://mlsanity.com/paper/2603.07108/) (2026)
- [Gaussian Process Inference Using Mini-batch Stochastic Gradient Descent: Convergence Guarantees and Empirical Benefits](https://mlsanity.com/paper/2111.10461/) (2021)
- [Modeling Massive Spatial Datasets Using a Conjugate Bayesian Linear Regression Framework](https://mlsanity.com/paper/2109.04447/) (2021)
- [A Spatio-Temporal Machine Learning Model for Mortgage Credit Risk: Default Probabilities and Loan Portfolios](https://mlsanity.com/paper/2410.02846/) (2024)
- [Scalable Model-Based Gaussian Process Clustering](https://mlsanity.com/paper/2309.07882/) (2023)
- [Implementation and Analysis of GPU Algorithms for Vecchia Approximation](https://mlsanity.com/paper/2407.02740/) (2024)
- [Spatial Process Approximations: Assessing Their Necessity](https://mlsanity.com/paper/2311.03201/) (2023)
- [Density correction for multivariate spatial fields of global climate model output using deep learning](https://mlsanity.com/paper/2411.18799/) (2024)
- [When the whole is greater than the sum of its parts: Scaling black-box inference to large data settings through divide-and-conquer](https://mlsanity.com/paper/2412.20323/) (2024)
- [DeepRV: Accelerating Spatiotemporal Inference with Pre-trained Neural Priors](https://mlsanity.com/paper/2503.21473/) (2025)
- [Modeling nonstationary spatial processes with normalizing flows](https://mlsanity.com/paper/2509.12884/) (2025)
- [A Streaming Sparse Cholesky Method for Derivative-Informed Gaussian Process Surrogates Within Digital Twin Applications](https://mlsanity.com/paper/2511.00366/) (2025)
- [Deep classifier kriging for probabilistic spatial prediction of air quality index](https://mlsanity.com/paper/2512.23474/) (2025)
- [GraphSVR: A Graph Convolutional Support Vector Regression Framework for Robust Spatiotemporal Air Pollution Forecasting](https://mlsanity.com/paper/2605.03795/) (2026)
- [GraphGP: Scalable Gaussian Processes with Vecchia's Approximation](https://mlsanity.com/paper/2606.11402/) (2026)
- [Gaussian Process Decorrelation for Spatiotemporal Deep Learning-Based Snow Water Equivalent Prediction](https://mlsanity.com/paper/2609.22182/) (2026)
- [Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matrices](https://mlsanity.com/paper/2609.30348/) (2026)
- [Scalable Sparse Autoregressive Variational Inference for Non-Gaussian Spatial Fields](https://mlsanity.com/paper/2610.13656/) (2026)

## Abstract (arXiv)

Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which can be viewed as a deficiency because the exact likelihood is permutation-invariant. This article takes the alternative standpoint that the ordering of the observations can be tuned to sharpen the approximations. Advantageously chosen orderings can drastically improve the approximations, and in fact, completely random orderings often produce far more accurate approximations than default coordinate-based orderings do. In addition to the permutation results, automatic methods for grouping calculations of components of the approximation are introduced, having the result of simultaneously improving the quality of the approximation and reducing its computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by a factor of 80 and computation time by a factor of 2 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equation approximations. Computational details are provided, including efficiently finding the orderings and ordered nearest neighbors, and profiling out linear mean parameters and using the approximations for prediction and conditional simulation. An application to space-time satellite data is presented.