Certifiable Estimation with Factor Graphs

2026-03-01

AI Summary

This work addresses the limitations of traditional factor graph estimation, which relies on local optimization and is prone to suboptimal solutions, thereby failing to meet the reliability demands of safety-critical applications requiring global optimality. While existing certifiable methods guarantee global optimality, their high computational complexity hinders practical deployment. The paper presents the first insight that the structure of factor graphs remains invariant under both Shor’s convex relaxation and Burer–Monteiro low-rank factorization. Leveraging this property, the authors develop a certifiable estimation framework compatible with mainstream factor graph libraries. By integrating QCQP modeling, Riemannian Staircase optimization, and native factor graph structure, the approach delivers verifiably globally optimal state estimates without altering existing workflows, significantly enhancing the practicality and deployability of certifiable estimation in real-world systems.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Factor graphs provide a convenient modular modeling language that enables practitioners to design and deploy high-performance robotic state estimation systems by composing simple, reusable building blocks. However, inference in these models is typically performed using local optimization methods that can converge to suboptimal solutions, a serious reliability concern in safety-critical applications. Conversely, certifiable estimators based on convex relaxation can recover verifiably globally optimal solutions in many practical settings, but the computational cost of solving their large-scale relaxations necessitates specialized, structure-exploiting solvers that require substantial expertise to implement, significantly hampering practical deployment. In this paper, we show that these two paradigms, which have thus far been treated as independent in the literature, can be naturally synthesized into a unified framework for certifiable factor graph optimization. The key insight is that factor graph structure is preserved under Shor's relaxation and Burer-Monteiro factorization: applying these transformations to a QCQP with an associated factor graph representation yields a lifted problem admitting a factor graph model with identical connectivity, in which variables and factors are simple one-to-one algebraic transformations of those in the original QCQP. This structural preservation enables the Riemannian Staircase methodology for certifiable estimation to be implemented using the same mature, highly-performant factor graph libraries and workflows already ubiquitously employed throughout robotics and computer vision, making certifiable estimation as straightforward to design and deploy as conventional factor graph inference.

Problem

• certifiable estimation
• factor graphs
• state estimation
• convex relaxation
• safety-critical applications

Innovation

• Factor Graphs
• Certifiable Estimation
• Shor's Relaxation
• Burer-Monteiro Factorization
• Riemannian Staircase