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

2026-03-17

AI Summary

This work addresses the limitations of multi-class Gaussian processes, which suffer from a lack of conjugacy and poor probability calibration. The authors propose a novel approach by leveraging Aitchison geometry, which provides a bijective mapping between the probability simplex and an unconstrained Euclidean space. This transformation recasts the multi-class classification problem as a low-dimensional Gaussian process regression, enabling conjugate inference without resorting to approximate posteriors. By integrating sparse GP techniques, the method achieves scalable inference while preserving well-calibrated predictive probabilities. Empirical evaluations on both synthetic and real-world datasets demonstrate that the model delivers competitive classification accuracy alongside highly reliable uncertainty estimates.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• multiclass Gaussian process
• probability calibration
• conjugate inference
• simplex geometry
• scalable classification

Innovation

• Simplex-to-Euclidean bijection
• Aitchison geometry
• Conjugate Gaussian process
• Calibrated classification
• Multi-class GP regression