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

Probabilistic inference over spatially embedded variables requires beliefs that respect $SE(3)$ symmetry, yet existing equivariant networks produce only scalars and vectors -- not the rank-2 precision tensors needed for anisotropic uncertainty, and single-component messages collapse multi-modal energy landscapes to physically meaningless averages. We introduce Equivariant Neural Belief Propagation (ENBP), a factor-graph framework whose messages are equivariant Gaussian mixture models with sufficient statistics that transform exactly under $SE(3)$. Rank-2 precision matrices are synthesised via equivariant outer products, ingested through differentiable spectral decomposition, and kept tractable by a greedy KL-based mixture reduction that provably commutes with $SE(3)$. On GEOM-QM9 and GEOM-Drugs, ENBP achieves 98.9% conformational coverage at 0.090 $\mathring{A}$ error with sub-second latency -- over $100\times$ faster than diffusion baselines at higher accuracy. On multi-body robotic inference, vanilla loopy BP diverges at 15+ agents while ENBP converges with near-zero collision rates and machine-precision equivariance error (${\sim}10^{-7}$ vs.\ $10^{-1}$ for augmented baselines).

Multi-robot patrolling requires a team to visit all areas of an environment at regular intervals, typically minimising idleness. A practical extension, motivated by security and environmental monitoring, is to additionally form a collective ranking of all patrol locations by some measured signal, a generalisation of the best-of-n problem to the many-option, continuous-valued regime. We observe that the patrol graph admits a natural dual interpretation: it is simultaneously the topology that dictates agent movement and a factor graph over which spatial beliefs can be propagated. Exploiting this equivalence, we apply Gaussian Belief Propagation (GBP), a graph-based algorithm, to collective ranking using unary measurement factors at visited nodes and pairwise smoothness factors along patrol edges. We compare GBP against simple and visit-count-weighted averaging across a range of sensor-noise conditions in simulation, and validate the approach on four Leo Rovers tracking a propagating radio signal in an office lobby. GBP outperforms both baselines on ranking accuracy, mean squared error, and time to consensus. We find that as noise increases and the task becomes harder, GBP degrades gracefully in simulation while both averaging methods degrade substantially. Hardware trials reproduce the same performance ordering on a real propagating radio signal, supporting the practical relevance of the simulated results.

Marginal inference in discrete graphical models forces a choice between exactness and scalability: exact algorithms are intractable for high-treewidth graphs, while iterative approximations (Belief Propagation, variational methods) sacrifice convergence guarantees on frustrated topologies. We argue that this dichotomy stems from a mismatched inductive bias: iterative methods abandon the sequential elimination structure that makes exact inference correct. We introduce In-Context Graphical Inference (ICG-I), an autoregressive Graph Transformer that restores this structure by mimicking Variable Elimination with learned, Tensor- Train-compressed intermediate factors, paired with a Dirichlet output layer and Weighted Conformal Prediction for calibrated, distribution-free coverage guarantees under topological shift. We prove that TT compression errors propagate at most lincarly through the autoregressive chain, that the Dirichlet-Multinomial loss is a proper scoring rule, and that WCP maintains coverage with a quantifiable degradation under estimated density ratios. We conducted intensive experiments to evaluate ICG-I and achieved state-of-the-art performance across all benchmarks. ICG-I reduces MAE from 0.041 (best baseline) to 0.020 on standard instances and achieves 0.048 on N=500 frustrated spin glasses where BP diverges entirely.