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Gaussian Process Kernels for Pattern Discovery and Extrapolation

arXiv 1302.4245 ↗First posted 18 Feb 2013stat.ML cs.AI stat.MEcited by 131

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Abstract (arXiv)

Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns and enable extrapolation. These kernels are derived by modelling a spectral density -- the Fourier transform of a kernel -- with a Gaussian mixture. The proposed kernels support a broad class of stationary covariances, but Gaussian process inference remains simple and analytic. We demonstrate the proposed kernels by discovering patterns and performing long range extrapolation on synthetic examples, as well as atmospheric CO2 trends and airline passenger data. We also show that we can reconstruct standard covariances within our framework.

Key facts. Cited by 131 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 Feb 2013.

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Citing papers per year: 2021: 29; 2022: 24; 2023: 18; 2024: 19; 2025: 20; 2026: 21.

131 papers.

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  1. 2023A decoder-only foundation model for time-series forecastingcited by 650
  2. 2023Large Language Models Are Zero-Shot Time Series Forecasterscited by 491
  3. 2021Priors in Bayesian Deep Learning: A Reviewcited by 101
  4. 2021Few-Shot Bayesian Optimization with Deep Kernel Surrogatescited by 70
  5. 2023The No Free Lunch Theorem, Kolmogorov Complexity, and the Role of Inductive Biases in Machine Learningcited by 63
  6. 2022Bayesian Model Selection, the Marginal Likelihood, and Generalizationcited by 56
  7. 2025Deep Learning is Not So Mysterious or Differentcited by 55
  8. 2021Benchmarking the Performance of Bayesian Optimization across Multiple Experimental Materials Science Domainscited by 46
  9. 2021Causal Inference for Time series Analysis: Problems, Methods and Evaluationcited by 42
  10. 2024Automated Statistical Model Discovery with Language Modelscited by 41
  11. 2021Deep Neural Networks as Point Estimates for Deep Gaussian Processescited by 26
  12. 2024Compute Better Spent: Replacing Dense Layers with Structured Matricescited by 24
38 more
  1. 2021Kernel Interpolation for Scalable Online Gaussian Processescited by 18
  2. 2021Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applicationscited by 17
  3. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  4. 2021High-Dimensional Gaussian Process Inference with Derivativescited by 16
  5. 2022Markovian Gaussian Process Variational Autoencoderscited by 16
  6. 2025Adaptive Kernel Design for Bayesian Optimization Is a Piece of CAKE with LLMscited by 15
  7. 2021On Learning the Transformer Kernelcited by 14
  8. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  9. 2023A spectrum of physics-informed Gaussian processes for regression in engineeringcited by 13
  10. 2021Non-Gaussian Gaussian Processes for Few-Shot Regressioncited by 12
  11. 2022Rethinking Bayesian Learning for Data Analysis: The Art of Prior and Inference in Sparsity-Aware Modelingcited by 12
  12. 2022Parallel MCMC Without Embarrassing Failurescited by 11
  13. 2022Scalable First-Order Bayesian Optimization via Structured Automatic Differentiationcited by 10
  14. 2024Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learningcited by 10
  15. 2023Multi-modal Gaussian Process Variational Autoencoders for Neural and Behavioral Datacited by 9
  16. 2021Scalable Variational Gaussian Processes via Harmonic Kernel Decompositioncited by 8
  17. 2022Nonstationary multi-output Gaussian processes via harmonizable spectral mixturescited by 8
  18. 2022Physics-informed machine learning for Structural Health Monitoringcited by 8
  19. 2022Structural Kernel Search via Bayesian Optimization and Symbolical Optimal Transportcited by 8
  20. 2021Correlated Product of Experts for Sparse Gaussian Process Regressioncited by 7
  21. 2024DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networkscited by 7
  22. 2021Measuring the robustness of Gaussian processes to kernel choicecited by 6
  23. 2021Faster Kernel Interpolation for Gaussian Processescited by 5
  24. 2021Recent Advances in Data-Driven Wireless Communication Using Gaussian Processes: A Comprehensive Surveycited by 5
  25. 2022Gaussian Process Surrogate Models for Neural Networkscited by 5
  26. 2023Discovery of structure-property relations for molecules via hypothesis-driven active learning over the chemical spacecited by 5
  27. 2023FaIRGP: A Bayesian Energy Balance Model for Surface Temperatures Emulationcited by 5
  28. 2024Extrapolation-Aware Nonparametric Statistical Inferencecited by 5
  29. 2024Dynamic Online Ensembles of Basis Expansionscited by 5
  30. 2021Learning Nonparametric Volterra Kernels with Gaussian Processescited by 4
  31. 2021Marginalising over Stationary Kernels with Bayesian Quadraturecited by 4
  32. 2021Hida-Mat'ern Kernelcited by 4
  33. 2021Self-supervised optimization of random material microstructures in the small-data regimecited by 4
  34. 2023Towards Flexibility and Interpretability of Gaussian Process State-Space Modelcited by 4
  35. 2023Bayesian Non-linear Latent Variable Modeling via Random Fourier Featurescited by 4
  36. 2024Learning Time-Varying Multi-Region Brain Communications via Scalable Markovian Gaussian Processescited by 4
  37. 2021Gaussian Process Nowcasting: Application to COVID-19 Mortality Reportingcited by 3
  38. 2021The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domaincited by 3

50 most cited of 131 All 131 (JSON)