# Sparse Variational Student-t Processes

arXiv: https://arxiv.org/abs/2312.05568 · HTML: https://mlsanity.com/paper/2312.05568/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 9 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Student-t Processes as Alternatives to Gaussian Processes](https://mlsanity.com/paper/1402.4306/) (2014)
- [Random Feature Expansions for Deep Gaussian Processes](https://mlsanity.com/paper/1610.04386/) (2016)
- [Inducing Point Allocation for Sparse Gaussian Processes in High-Throughput Bayesian Optimisation](https://mlsanity.com/paper/2301.10123/) (2023)
- [Deep convolutional Gaussian processes](https://mlsanity.com/paper/1810.03052/) (2018)
- [Multivariate Gaussian and Student-t Process Regression for Multi-output Prediction](https://mlsanity.com/paper/1703.04455/) (2017)
- [Trust Your Robots! Predictive Uncertainty Estimation of Neural Networks with Sparse Gaussian Processes](https://mlsanity.com/paper/2109.09690/) (2021)
- [Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula](https://mlsanity.com/paper/1701.05638/) (2017)
- [The Elliptical Processes: a Family of Fat-tailed Stochastic Processes](https://mlsanity.com/paper/2003.07201/) (2020)

## Built on by (most cited first, 8 of 8)

- [Diffusion Bridge Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/2509.19078/) (2025)
- [A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2602.00834/) (2026)
- [Sparse Variational Student-t Processes for Heavy-tailed Modeling](https://mlsanity.com/paper/2408.06699/) (2024)
- [Towards Disentangled Preference Optimization Dynamics: Suppress the Loser, Preserve the Winner](https://mlsanity.com/paper/2604.18239/) (2026)
- [Multivariate Bayesian Last Layer for Regression with Uncertainty Quantification and Decomposition](https://mlsanity.com/paper/2405.01761/) (2024)
- [Neural Bridge Processes](https://mlsanity.com/paper/2508.07220/) (2025)
- [Implicit Variational Rejection Sampling](https://mlsanity.com/paper/2606.14235/) (2026)
- [DP-Splat: Bayesian Nonparametric Complexity Control for Gaussian Splatting](https://mlsanity.com/paper/2607.10912/) (2026)

## Abstract (arXiv)

The theory of Bayesian learning incorporates the use of Student-t Processes to model heavy-tailed distributions and datasets with outliers. However, despite Student-t Processes having a similar computational complexity as Gaussian Processes, there has been limited emphasis on the sparse representation of this model. This is mainly due to the increased difficulty in modeling and computation compared to previous sparse Gaussian Processes. Our motivation is to address the need for a sparse representation framework that reduces computational complexity, allowing Student-t Processes to be more flexible for real-world datasets. To achieve this, we leverage the conditional distribution of Student-t Processes to introduce sparse inducing points. Bayesian methods and variational inference are then utilized to derive a well-defined lower bound, facilitating more efficient optimization of our model through stochastic gradient descent. We propose two methods for computing the variational lower bound, one utilizing Monte Carlo sampling and the other employing Jensen's inequality to compute the KL regularization term in the loss function. We propose adopting these approaches as viable alternatives to Gaussian processes when the data might contain outliers or exhibit heavy-tailed behavior, and we provide specific recommendations for their applicability. We evaluate the two proposed approaches on various synthetic and real-world datasets from UCI and Kaggle, demonstrating their effectiveness compared to baseline methods in terms of computational complexity and accuracy, as well as their robustness to outliers.