# Variational Gaussian Processes For Linear Inverse Problems

arXiv: https://arxiv.org/abs/2311.00663 · HTML: https://mlsanity.com/paper/2311.00663/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 78 references that are in the map.
- First posted on arXiv 1 Nov 2023.

## Builds on (most cited first, 11 of 11)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Rates of Convergence for Sparse Variational Gaussian Process Regression](https://mlsanity.com/paper/1903.03571/) (2019)
- [Frequentist Consistency of Variational Bayes](https://mlsanity.com/paper/1705.03439/) (2017)
- [On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes](https://mlsanity.com/paper/1504.07027/) (2015)
- [Convergence Rates of Variational Posterior Distributions](https://mlsanity.com/paper/1712.02519/) (2017)
- [Variational Bayes for high-dimensional linear regression with sparse priors](https://mlsanity.com/paper/1904.07150/) (2019)
- [Improved Convergence Rates for Sparse Approximation Methods in Kernel-Based Learning](https://mlsanity.com/paper/2202.04005/) (2022)
- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [A variational inference framework for inverse problems](https://mlsanity.com/paper/2103.05909/) (2021)
- [Pointwise uncertainty quantification for sparse variational Gaussian process regression with a Brownian motion prior](https://mlsanity.com/paper/2310.00097/) (2023)

## Built on by (most cited first, 2 of 2)

- [Adaptive sparse variational approximations for Gaussian process regression](https://mlsanity.com/paper/2504.03321/) (2025)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)

## Abstract (arXiv)

By now Bayesian methods are routinely used in practice for solving inverse problems. In inverse problems the parameter or signal of interest is observed only indirectly, as an image of a given map, and the observations are typically further corrupted with noise. Bayes offers a natural way to regularize these problems via the prior distribution and provides a probabilistic solution, quantifying the remaining uncertainty in the problem. However, the computational costs of standard, sampling based Bayesian approaches can be overly large in such complex models. Therefore, in practice variational Bayes is becoming increasingly popular. Nevertheless, the theoretical understanding of these methods is still relatively limited, especially in context of inverse problems. In our analysis we investigate variational Bayesian methods for Gaussian process priors to solve linear inverse problems. We consider both mildly and severely ill-posed inverse problems and work with the popular inducing variables variational Bayes approach proposed by Titsias in 2009. We derive posterior contraction rates for the variational posterior in general settings and show that the minimax estimation rate can be attained by correctly tunned procedures. As specific examples we consider a collection of inverse problems including the heat equation, Volterra operator and Radon transform and inducing variable methods based on population and empirical spectral features.