# Sparse Algorithms for Markovian Gaussian Processes

arXiv: https://arxiv.org/abs/2103.10710 · HTML: https://mlsanity.com/paper/2103.10710/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 47 references that are in the map.
- First posted on arXiv 19 Mar 2021.

## Builds on (most cited first, 21 of 21)

- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Conjugate-Computation Variational Inference : Converting Variational Inference in Non-Conjugate Models to Inferences in Conjugate Models](https://mlsanity.com/paper/1703.04265/) (2017)
- [Rates of Convergence for Sparse Variational Gaussian Process Regression](https://mlsanity.com/paper/1903.03571/) (2019)
- [Sparse Gaussian Processes with Spherical Harmonic Features](https://mlsanity.com/paper/2006.16649/) (2020)
- [Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Models](https://mlsanity.com/paper/1803.09151/) (2018)
- [Stochastic Expectation Propagation](https://mlsanity.com/paper/1506.04132/) (2015)
- [Fast Variational Learning in State-Space Gaussian Process Models](https://mlsanity.com/paper/2007.04731/) (2020)
- [Faster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions](https://mlsanity.com/paper/1511.00146/) (2015)
- [Doubly Sparse Variational Gaussian Processes](https://mlsanity.com/paper/2001.05363/) (2020)
- [State Space Gaussian Processes with Non-Gaussian Likelihood](https://mlsanity.com/paper/1802.04846/) (2018)
- [Banded Matrix Operators for Gaussian Markov Models in the Automatic Differentiation Era](https://mlsanity.com/paper/1902.10078/) (2019)
- [State Space Expectation Propagation: Efficient Inference Schemes for Temporal Gaussian Processes](https://mlsanity.com/paper/2007.05994/) (2020)
- [Approximate Inference for Nonstationary Heteroscedastic Gaussian process Regression](https://mlsanity.com/paper/1404.5443/) (2014)
- [Infinite-Horizon Gaussian Processes](https://mlsanity.com/paper/1811.06588/) (2018)
- [Gaussian process classification using posterior linearisation](https://mlsanity.com/paper/1809.04967/) (2018)
- [End-to-End Probabilistic Inference for Nonstationary Audio Analysis](https://mlsanity.com/paper/1901.11436/) (2019)

## Built on by (most cited first, 10 of 10)

- [Spatio-Temporal Variational Gaussian Processes](https://mlsanity.com/paper/2111.01732/) (2021)
- [Dual Parameterization of Sparse Variational Gaussian Processes](https://mlsanity.com/paper/2111.03412/) (2021)
- [Markovian Gaussian Process Variational Autoencoders](https://mlsanity.com/paper/2207.05543/) (2022)
- [Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees](https://mlsanity.com/paper/2111.01721/) (2021)
- [Variational Gaussian Process Diffusion Processes](https://mlsanity.com/paper/2306.02066/) (2023)
- [Linear Time GPs for Inferring Latent Trajectories from Neural Spike Trains](https://mlsanity.com/paper/2306.01802/) (2023)
- [Recurrent Memory for Online Interdomain Gaussian Processes](https://mlsanity.com/paper/2502.08736/) (2025)
- [Spatiotemporal modeling of European paleoclimate using doubly sparse Gaussian processes](https://mlsanity.com/paper/2211.08160/) (2022)
- [Deep Gaussian Markov Random Fields for Graph-Structured Dynamical Systems](https://mlsanity.com/paper/2306.08445/) (2023)
- [Designing an Optimal Sensor Network via Minimizing Information Loss](https://mlsanity.com/paper/2512.05940/) (2025)

## Abstract (arXiv)

Approximate Bayesian inference methods that scale to very large datasets are crucial in leveraging probabilistic models for real-world time series. Sparse Markovian Gaussian processes combine the use of inducing variables with efficient Kalman filter-like recursions, resulting in algorithms whose computational and memory requirements scale linearly in the number of inducing points, whilst also enabling parallel parameter updates and stochastic optimisation. Under this paradigm, we derive a general site-based approach to approximate inference, whereby we approximate the non-Gaussian likelihood with local Gaussian terms, called sites. Our approach results in a suite of novel sparse extensions to algorithms from both the machine learning and signal processing literature, including variational inference, expectation propagation, and the classical nonlinear Kalman smoothers. The derived methods are suited to large time series, and we also demonstrate their applicability to spatio-temporal data, where the model has separate inducing points in both time and space.