Variable Elimination in Hybrid Factor Graphs for Discrete-Continuous Inference & Estimation

2026-01-02

arXiv.org

AI Summary

This work addresses the challenge of hybrid estimation in robotics, where discrete and continuous variables coexist, a setting in which traditional methods rely on approximations that hinder exact inference. The authors propose an efficient hybrid factor graph framework that introduces novel mixed Gaussian factors and hybrid conditional representations, leveraging conditional linear Gaussian mechanisms for variable elimination. This approach enables, for the first time, exact posterior inference within hybrid factor graphs, supporting both exact maximum a posteriori (MAP) estimation and marginalization. By exploiting a tree-structured factorization and a pruning-based assignment strategy, the method effectively controls the combinatorial growth of discrete hypotheses, achieving scalability without sacrificing accuracy. Experiments on SLAM datasets with ambiguous observations demonstrate the method’s accuracy, generality, and elegance in jointly estimating discrete measurement associations and continuous robot states.

Technology Category

• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

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

Abstract

Many hybrid problems in robotics involve both continuous and discrete components, and modeling them together for estimation tasks has been a long standing and difficult problem. Hybrid Factor Graphs give us a mathematical framework to model these types of problems, however existing approaches for solving them are based on approximations. In this work, we propose an efficient Hybrid Factor Graph framework alongwith a variable elimination algorithm to produce a hybrid Bayes network, which can then be used for exact Maximum A Posteriori estimation and marginalization over both sets of variables. Our approach first develops a novel hybrid Gaussian factor which can connect to both discrete and continuous variables, and a hybrid conditional which can represent multiple continuous hypotheses conditioned on the discrete variables. Using these representations, we derive the process of hybrid variable elimination under the Conditional Linear Gaussian scheme, giving us exact posteriors as hybrid Bayes network. To bound the number of discrete hypotheses, we use a tree-structured representation of the factors coupled with a simple pruning and probabilistic assignment scheme, which allows for tractable inference. We demonstrate the applicability of our framework on a SLAM dataset with ambiguous measurements, where discrete choices for the most likely measurement have to be made. Our demonstrated results showcase the accuracy, generality, and simplicity of our hybrid factor graph framework.

Problem

• hybrid inference
• discrete-continuous estimation
• variable elimination
• factor graphs
• Bayesian estimation

Innovation

• Hybrid Factor Graphs
• Variable Elimination
• Discrete-Continuous Inference
• Conditional Linear Gaussian
• Exact MAP Estimation