# Transforming Gaussian Processes With Normalizing Flows

arXiv: https://arxiv.org/abs/2011.01596 · HTML: https://mlsanity.com/paper/2011.01596/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2020.

## Built on by (most cited first, 28 of 28)

- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
- [Autoregressive Conditional Neural Processes](https://mlsanity.com/paper/2303.14468/) (2023)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Posterior Refinement Improves Sample Efficiency in Bayesian Neural Networks](https://mlsanity.com/paper/2205.10041/) (2022)
- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Non-Gaussian Gaussian Processes for Few-Shot Regression](https://mlsanity.com/paper/2110.13561/) (2021)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [Efficient Transformed Gaussian Processes for Non-Stationary Dependent Multi-class Classification](https://mlsanity.com/paper/2205.15008/) (2022)
- [Deep Stochastic Processes via Functional Markov Transition Operators](https://mlsanity.com/paper/2305.15574/) (2023)
- [Bayesian Gaussian Process ODEs via Double Normalizing Flows](https://mlsanity.com/paper/2309.09222/) (2023)
- [Towards Flexibility and Interpretability of Gaussian Process State-Space Model](https://mlsanity.com/paper/2301.08843/) (2023)
- [AdaAnn: Adaptive Annealing Scheduler for Probability Density Approximation](https://mlsanity.com/paper/2202.00792/) (2022)
- [Sample-Efficient Optimisation with Probabilistic Transformer Surrogates](https://mlsanity.com/paper/2205.13902/) (2022)
- [Towards Efficient Modeling and Inference in Multi-Dimensional Gaussian Process State-Space Models](https://mlsanity.com/paper/2309.01074/) (2023)
- [Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2503.18309/) (2025)
- [TreeFlow: Going beyond Tree-based Gaussian Probabilistic Regression](https://mlsanity.com/paper/2206.04140/) (2022)
- [Deep Transformed Gaussian Processes](https://mlsanity.com/paper/2310.18230/) (2023)
- [Variational Elliptical Processes](https://mlsanity.com/paper/2311.12566/) (2023)
- [GGMPs: Generalized Gaussian Mixture Processes](https://mlsanity.com/paper/2603.10442/) (2026)
- [Flow-Transformed Implicit Processes for Function-Space Variational Inference](https://mlsanity.com/paper/2606.01954/) (2026)
- [Generative structured normalizing flow Gaussian processes applied to spectroscopic data](https://mlsanity.com/paper/2212.07554/) (2022)
- [Applications of Gaussian Processes at Extreme Lengthscales: From Molecules to Black Holes](https://mlsanity.com/paper/2303.14291/) (2023)
- [Monte Carlo inference for semiparametric Bayesian regression](https://mlsanity.com/paper/2306.05498/) (2023)
- [Kernelised Normalising Flows](https://mlsanity.com/paper/2307.14839/) (2023)
- [Estimation of Counterfactual Interventions under Uncertainties](https://mlsanity.com/paper/2309.08332/) (2023)
- [Efficient modeling of sub-kilometer surface wind with Gaussian processes and neural networks](https://mlsanity.com/paper/2405.12614/) (2024)
- [Complex normalizing flows can almost be information K"ahler-Ricci flows](https://mlsanity.com/paper/2604.17954/) (2026)

## Abstract (arXiv)

Gaussian Processes (GPs) can be used as flexible, non-parametric function priors. Inspired by the growing body of work on Normalizing Flows, we enlarge this class of priors through a parametric invertible transformation that can be made input-dependent. Doing so also allows us to encode interpretable prior knowledge (e.g., boundedness constraints). We derive a variational approximation to the resulting Bayesian inference problem, which is as fast as stochastic variational GP regression (Hensman et al., 2013; Dezfouli and Bonilla,2015). This makes the model a computationally efficient alternative to other hierarchical extensions of GP priors (Lazaro-Gredilla,2012; Damianou and Lawrence, 2013). The resulting algorithm's computational and inferential performance is excellent, and we demonstrate this on a range of data sets. For example, even with only 5 inducing points and an input-dependent flow, our method is consistently competitive with a standard sparse GP fitted using 100 inducing points.