# Simplex-to-Euclidean Bijection for Conjugate and Calibrated Multiclass Gaussian Process Classification

arXiv: https://arxiv.org/abs/2603.16621 · HTML: https://mlsanity.com/paper/2603.16621/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 44 references that are in the map.
- First posted on arXiv 17 Mar 2026.

## Builds on (most cited first, 29 of 29)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Exact Gaussian Processes on a Million Data Points](https://mlsanity.com/paper/1903.08114/) (2019)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Dirichlet-based Gaussian Processes for Large-scale Calibrated Classification](https://mlsanity.com/paper/1805.10915/) (2018)
- [MCMC for Variationally Sparse Gaussian Processes](https://mlsanity.com/paper/1506.04000/) (2015)
- [Product Kernel Interpolation for Scalable Gaussian Processes](https://mlsanity.com/paper/1802.08903/) (2018)
- [GP-Tree: A Gaussian Process Classifier for Few-Shot Incremental Learning](https://mlsanity.com/paper/2102.07868/) (2021)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [Preconditioning for Scalable Gaussian Process Hyperparameter Optimization](https://mlsanity.com/paper/2107.00243/) (2021)
- [Deep convolutional Gaussian processes](https://mlsanity.com/paper/1810.03052/) (2018)
- [Bayesian Image Classification with Deep Convolutional Gaussian Processes](https://mlsanity.com/paper/1902.05888/) (2019)
- [Orthogonally Decoupled Variational Gaussian Processes](https://mlsanity.com/paper/1809.08820/) (2018)
- [Efficient Gaussian Process Classification Using Polya-Gamma Data Augmentation](https://mlsanity.com/paper/1802.06383/) (2018)
- [Stochastic Gradient Descent for Gaussian Processes Done Right](https://mlsanity.com/paper/2310.20581/) (2023)
- [Multi-Class Gaussian Process Classification Made Conjugate: Efficient Inference via Data Augmentation](https://mlsanity.com/paper/1905.09670/) (2019)
- [Adversarial Robustness Guarantees for Classification with Gaussian Processes](https://mlsanity.com/paper/1905.11876/) (2019)
- [Multi-class Gaussian Process Classification with Noisy Inputs](https://mlsanity.com/paper/2001.10523/) (2020)
- [Scalable Gaussian Process Classification via Expectation Propagation](https://mlsanity.com/paper/1507.04513/) (2015)
- [Efficient Transformed Gaussian Processes for Non-Stationary Dependent Multi-class Classification](https://mlsanity.com/paper/2205.15008/) (2022)
- [Scalable Multi-Class Gaussian Process Classification using Expectation Propagation](https://mlsanity.com/paper/1706.07258/) (2017)
- [SketchDNN: Joint Continuous-Discrete Diffusion for CAD Sketch Generation](https://mlsanity.com/paper/2507.11579/) (2025)
- [Flows on convex polytopes](https://mlsanity.com/paper/2503.10232/) (2025)
- [Simplex-to-Euclidean Bijections for Categorical Flow Matching](https://mlsanity.com/paper/2510.27480/) (2025)
- [Scalable Gaussian Process Classification with Additive Noise for Various Likelihoods](https://mlsanity.com/paper/1909.06541/) (2019)

## Abstract (arXiv)

We propose a conjugate and calibrated Gaussian process (GP) model for multi-class classification by exploiting the geometry of the probability simplex. Our approach uses Aitchison geometry to map simplex-valued class probabilities to an unconstrained Euclidean representation, turning classification into a GP regression problem with fewer latent dimensions than standard multi-class GP classifiers. This yields conjugate inference and reliable predictive probabilities without relying on distributional approximations in the model construction. The method is compatible with standard sparse GP regression techniques, enabling scalable inference on larger datasets. Empirical results show well-calibrated and competitive performance across synthetic and real-world datasets.