# Hierarchy-GBP: Accelerating Factor Graph Inference via Abstraction and Recovery

arXiv: https://arxiv.org/abs/2610.06978 · HTML: https://mlsanity.com/paper/2610.06978/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 45 references that are in the map.
- First posted on arXiv 4 Oct 2026.

## Builds on (most cited first, 12 of 12)

- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [FutureMapping: The Computational Structure of Spatial AI Systems](https://mlsanity.com/paper/1803.11288/) (2018)
- [FutureMapping 2: Gaussian Belief Propagation for Spatial AI](https://mlsanity.com/paper/1910.14139/) (2019)
- [Bundle Adjustment on a Graph Processor](https://mlsanity.com/paper/2003.03134/) (2020)
- [Square Root Bundle Adjustment for Large-Scale Reconstruction](https://mlsanity.com/paper/2103.01843/) (2021)
- [A visual introduction to Gaussian Belief Propagation](https://mlsanity.com/paper/2107.02308/) (2021)
- [Convergence Analysis of Distributed Inference with Vector-Valued Gaussian Belief Propagation](https://mlsanity.com/paper/1611.02010/) (2016)
- [Majorization Minimization Methods for Distributed Pose Graph Optimization](https://mlsanity.com/paper/2108.00083/) (2021)
- [Distributed Simultaneous Localisation and Auto-Calibration using Gaussian Belief Propagation](https://mlsanity.com/paper/2401.15036/) (2024)
- [Sparse Pose Graph Optimization in Cycle Space](https://mlsanity.com/paper/2203.15597/) (2022)
- [Hyperion -- A fast, versatile symbolic Gaussian Belief Propagation framework for Continuous-Time SLAM](https://mlsanity.com/paper/2407.07074/) (2024)
- [Belief Propagation Converges to Gaussian Distributions in Sparsely-Connected Factor Graphs](https://mlsanity.com/paper/2601.21935/) (2026)

## Abstract (arXiv)

Gaussian Belief Propagation (GBP) is a distributed inference algorithm that passes messages in graphical models, making it attractive for scalable spatial intelligence. However, we find GBP most effective locally: it rapidly smooths message errors that vary sharply between neighbor variables, but corrects global errors across distant graph regions incrementally through long-range message propagations. We propose Hierarchy-GBP (H-GBP), an iterative, two-stage framework that accelerates GBP by first solving these global errors with a coarse graph approximation (abstraction) and projecting the results back to the original graph (recovery), then refining the remaining local errors with GBP. We prove H-GBP convergence to the optimum by deriving the combined matrix operator of our abstraction and recovery steps and analyzing its spectral radius. Experiments on linear sparse graphs show that H-GBP converges fundamentally faster than standard GBP. Moreover, we validate H-GBP on two important spatial problems: Pose Graph Optimization (PGO) and Bundle Adjustment (BA). H-GBP markedly accelerates large-scale PGO and achieves state-of-the-art runtime across all tested BA scales.