# The Variational Gaussian Process

arXiv: https://arxiv.org/abs/1511.06499 · HTML: https://mlsanity.com/paper/1511.06499/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2015.

## Built on by (most cited first, 30 of 30)

- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [DIME:Diffusion-Based Maximum Entropy Reinforcement Learning](https://mlsanity.com/paper/2502.02316/) (2025)
- [Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks](https://mlsanity.com/paper/2110.13330/) (2021)
- [Non-Adversarial Imitation Learning and its Connections to Adversarial Methods](https://mlsanity.com/paper/2008.03525/) (2020)
- [Cauchy-Schwarz Regularized Autoencoder](https://mlsanity.com/paper/2101.02149/) (2021)
- [Efficient Semi-Implicit Variational Inference](https://mlsanity.com/paper/2101.06070/) (2021)
- [MuyGPs: Scalable Gaussian Process Hyperparameter Estimation Using Local Cross-Validation](https://mlsanity.com/paper/2104.14581/) (2021)
- [Variational Laplace Autoencoders](https://mlsanity.com/paper/2211.17267/) (2022)
- [Reducing the Amortization Gap in Variational Autoencoders: A Bayesian Random Function Approach](https://mlsanity.com/paper/2102.03151/) (2021)
- [Variational Marginal Particle Filters](https://mlsanity.com/paper/2109.15134/) (2021)
- [A Sparse Expansion For Deep Gaussian Processes](https://mlsanity.com/paper/2112.05888/) (2021)
- [A Survey on Uncertainty Toolkits for Deep Learning](https://mlsanity.com/paper/2205.01040/) (2022)
- [Calibrated simplex-mapping classification](https://mlsanity.com/paper/2103.02926/) (2021)
- [Non-separable Covariance Kernels for Spatiotemporal Gaussian Processes based on a Hybrid Spectral Method and the Harmonic Oscillator](https://mlsanity.com/paper/2302.09580/) (2023)
- [On the Size and Approximation Error of Distilled Sets](https://mlsanity.com/paper/2305.14113/) (2023)
- [Efficient and Principled Scientific Discovery through Bayesian Optimization: A Tutorial](https://mlsanity.com/paper/2604.01328/) (2026)
- [A Self-Supervised Framework for Function Learning and Extrapolation](https://mlsanity.com/paper/2106.07369/) (2021)
- [Scalable Safety-Critical Policy Evaluation with Accelerated Rare Event Sampling](https://mlsanity.com/paper/2106.10566/) (2021)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (2022)
- [VFDS: Variational Foresight Dynamic Selection in Bayesian Neural Networks for Efficient Human Activity Recognition](https://mlsanity.com/paper/2204.00130/) (2022)
- [Fast Gaussian Process Posterior Mean Prediction via Local Cross Validation and Precomputation](https://mlsanity.com/paper/2205.10879/) (2022)
- [Latent Variable Modelling Using Variational Autoencoders: A survey](https://mlsanity.com/paper/2206.09891/) (2022)
- [Structured Recognition for Generative Models with Explaining Away](https://mlsanity.com/paper/2209.05212/) (2022)
- [Variational Elliptical Processes](https://mlsanity.com/paper/2311.12566/) (2023)
- [A Metalearned Neural Circuit for Nonparametric Bayesian Inference](https://mlsanity.com/paper/2311.14601/) (2023)
- [Algorithmic Identification of Essential Exogenous Nodes for Causal Sufficiency in Brain Networks](https://mlsanity.com/paper/2403.05407/) (2024)
- [SEED-SET: Scalable Evolving Experimental Design for System-level Ethical Testing](https://mlsanity.com/paper/2603.01630/) (2026)
- [Variational Boosted Soft Trees](https://mlsanity.com/paper/2302.10706/) (2023)
- [Decentralized Indoor Localization Based on A Sparse Gaussian Process with Reduced-Dimensional Inputs for Real-Time Sensing and Training on IoT Devices](https://mlsanity.com/paper/2409.00078/) (2024)
- [(MPO)2: Multivariate Polynomial Optimization based on Matrix Product Operators](https://mlsanity.com/paper/2607.15916/) (2026)

## Abstract (arXiv)

Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We develop the variational Gaussian process (VGP), a Bayesian nonparametric variational family, which adapts its shape to match complex posterior distributions. The VGP generates approximate posterior samples by generating latent inputs and warping them through random non-linear mappings; the distribution over random mappings is learned during inference, enabling the transformed outputs to adapt to varying complexity. We prove a universal approximation theorem for the VGP, demonstrating its representative power for learning any model. For inference we present a variational objective inspired by auto-encoders and perform black box inference over a wide class of models. The VGP achieves new state-of-the-art results for unsupervised learning, inferring models such as the deep latent Gaussian model and the recently proposed DRAW.