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

TL;DR

This paper introduces a novel Gaussian process model for multi-class classification that leverages Aitchison geometry to map class probabilities from the simplex to Euclidean space, enabling scalable, conjugate inference with well-calibrated predictions.

Contribution

It presents a new simplex-to-Euclidean bijection for multi-class GPs, improving calibration and scalability without distributional approximations.

Findings

Achieves well-calibrated predictive probabilities.

Demonstrates competitive performance on real-world datasets.

Enables scalable inference with standard sparse GP techniques.

Abstract

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.