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A general framework for Vecchia approximations of Gaussian processes

arXiv 1708.06302 ↗First posted 21 Aug 2017stat.ME stat.COcited by 66

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Abstract (arXiv)

Gaussian processes (GPs) are commonly used as models for functions, time series, and spatial fields, but they are computationally infeasible for large datasets. Focusing on the typical setting of modeling data as a GP plus an additive noise term, we propose a generalization of the Vecchia (1988) approach as a framework for GP approximations. We show that our general Vecchia approach contains many popular existing GP approximations as special cases, allowing for comparisons among the different methods within a unified framework. Representing the models by directed acyclic graphs, we determine the sparsity of the matrices necessary for inference, which leads to new insights regarding the computational properties. Based on these results, we propose a novel sparse general Vecchia approximation, which ensures computational feasibility for large spatial datasets but can lead to considerable improvements in approximation accuracy over Vecchia's original approach. We provide several theoretical results and conduct numerical comparisons. We conclude with guidelines for the use of Vecchia approximations in spatial statistics.

Key facts. Cited by 66 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 Aug 2017.

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Citing papers per year: 2021: 6; 2022: 7; 2023: 11; 2024: 15; 2025: 13; 2026: 14.

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  1. 2024Neural Methods for Amortized Inferencecited by 36
  2. 2022Posterior and Computational Uncertainty in Gaussian Processescited by 23
  3. 2021Bias-Free Scalable Gaussian Processes via Randomized Truncationscited by 15
  4. 2022Scalable Bayesian Optimization Using Vecchia Approximations of Gaussian Processescited by 11
  5. 2023Large-Scale Gaussian Processes via Alternating Projectioncited by 11
  6. 2023Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoodscited by 10
  7. 2022Exact Gaussian Processes for Massive Datasets via Non-Stationary Sparsity-Discovering Kernelscited by 9
  8. 2022Variational Nearest Neighbor Gaussian Processcited by 8
  9. 2022Scalable Gaussian-process regression and variable selection using Vecchia approximationscited by 8
  10. 2024Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inferencecited by 8
  11. 2021Correlated Product of Experts for Sparse Gaussian Process Regressioncited by 7
  12. 2023CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebracited by 7
38 more
  1. 2021Scalable Cross Validation Losses for Gaussian Process Modelscited by 6
  2. 2023Iterative Methods for Vecchia-Laplace Approximations for Latent Gaussian Process Modelscited by 6
  3. 2021A Sparse Expansion For Deep Gaussian Processescited by 5
  4. 2022Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Treescited by 5
  5. 2023Sharded Bayesian Additive Regression Treescited by 5
  6. 2023Variational sparse inverse Cholesky approximation for latent Gaussian processes via double Kullback-Leibler minimizationcited by 4
  7. 2024Iterative Methods for Full-Scale Gaussian Process Approximations for Large Spatial Datacited by 4
  8. 2021Learning particle swarming models from data with Gaussian processescited by 3
  9. 2023Vecchia Gaussian Process Ensembles on Internal Representations of Deep Neural Networkscited by 3
  10. 2025dynoGP: Deep Gaussian Processes for dynamic system identificationcited by 3
  11. 2025Neighbour-Driven Gaussian Process Variational Autoencoders for Scalable Structured Latent Modellingcited by 3
  12. 2025All Emulators are Wrong, Many are Useful, and Some are More Useful Than Others: A Reproducible Comparison of Computer Model Surrogatescited by 3
  13. 2021Modeling Massive Spatial Datasets Using a Conjugate Bayesian Linear Regression Frameworkcited by 2
  14. 2024Posterior Covariance Structures in Gaussian Processescited by 2
  15. 2024A Spatio-Temporal Machine Learning Model for Mortgage Credit Risk: Default Probabilities and Loan Portfolioscited by 2
  16. 2024Conjugate gradient methods for high-dimensional GLMMscited by 2
  17. 2024Compactly-supported nonstationary kernels for computing exact Gaussian processes on big datacited by 2
  18. 2025Bayesian Data Sketching for Varying Coefficient Regression Modelscited by 2
  19. 2022Dynamic Bayesian Learning for Spatiotemporal Mechanistic Modelscited by 1
  20. 2023Scalable Model-Based Gaussian Process Clusteringcited by 1
  21. 2023Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regressioncited by 1
  22. 2024Implementation and Analysis of GPU Algorithms for Vecchia Approximationcited by 1
  23. 2024Probabilistic Spatiotemporal Modeling of Day-Ahead Wind Power Generation with Input-Warped Gaussian Processescited by 1
  24. 2024Positional Encoder Graph Quantile Neural Networks for Geographic Datacited by 1
  25. 2024Learning non-Gaussian spatial distributions via Bayesian transport maps with parametric shrinkagecited by 1
  26. 2025Predictive posterior sampling from non-stationnary Gaussian process priors via Diffusion models with application to climate datacited by 1
  27. 2025gp2Scale: A Class of Compactly Supported Non-Stationary Kernels and Distributed Computing for Exact Gaussian Processes on 10 Million Data Pointscited by 1
  28. 2026Spatio-temporal modeling and forecasting with Fourier neural operatorscited by 1
  29. 2026Gradient-based Active Learning with Gaussian Processes for Global Sensitivity Analysiscited by 1
  30. 2023Privacy-aware Gaussian Process Regressioncited by 0
  31. 2023Monte Carlo inference for semiparametric Bayesian regressioncited by 0
  32. 2024Beyond State Space Representation: A General Theory for Kernel Packetscited by 0
  33. 2024Gaussian Processes Sampling with Sparse Grids under Additive Schwarz Preconditionercited by 0
  34. 2024Aggregation Models with Optimal Weights for Distributed Gaussian Processescited by 0
  35. 2024Density correction for multivariate spatial fields of global climate model output using deep learningcited by 0
  36. 2025DeepRV: Accelerating Spatiotemporal Inference with Pre-trained Neural Priorscited by 0
  37. 2025Vecchia-Inducing-Points Full-Scale Approximations for Gaussian Processescited by 0
  38. 2025Fast Gaussian process inference by exact Mat'ern kernel decompositioncited by 0

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