Abstract
Belief Propagation (BP) has operated in Euclidean space for four decades, yet the polynomial volume growth of ℝ^d is fundamentally mismatched to hierarchical graphs: faithful embedding of exponentially branching structure demands d = Ω(log n) dimensions and prohibitive 𝒪(d^3) covariance costs, while truncating d corrupts marginal estimates. We introduce Continuous Hyperbolic Belief Propagation (CHBP), formulated natively on the Lorentz hyperboloid ℍ^n_K, whose exponential volume growth eliminates this bottleneck. CHBP parameterises beliefs as Jacobian-corrected Wrapped Normals, approximates message integrals via Gauss–Hermite quadrature, and transports covariance tensors between tangent spaces via closed-form Levi-Civita parallel transport, with a curvature annealing schedule ensuring stable convergence. On hierarchical graphs, CHBP at d=5 reduces marginal KL divergence by 25× versus Euclidean BP at d=50 and achieves up to 91.45