HyRef-SB: Hyperbolic Schrödinger Bridges Guided by Geometric References

Endpoint distributions specify where stochastic transport starts and ends, while reference dynamics provide a geometric inductive bias for the resulting stochastic paths between them. We introduce Hyperbolic Reference-guided Schrödinger Bridges (HyRef-SB), combining reference diffusion, stochastic control, and endpoint distribution constraints on a hyperbolic manifold. An invertible calibration preserves the input representation, while a prescribed base metric defines diffusion and control costs. An endpoint-preserving path teacher calibrates the reference drift; differentiable rollouts then jointly minimize intrinsic residual control energy and a soft endpoint penalty under the fixed reference. A shared backbone and conditional adapters model multiple terminal distributions. Our analysis characterizes the radial and transverse costs induced by calibration and quantifies, for regular teacher marginals, how reference regression error and density regularity bound reference marginal error and a feasible residual-control budget. Continuous and finite-step entropy identities identify the regularized path laws, and a population penalty limit connects soft endpoint fitting to hard marginal constraints. Experiments yield low-collision navigation and recover tree-structure statistics, while improving cellular distribution matching over the evaluated baselines on most terminal and held-out intermediate metrics. Hyperbolic geometry reduces terminal errors relative to scale-matched Euclidean controls on three cellular tasks, and the Trametinib ablation shows further gains from the learned reference over a Brownian reference. The complete code will be made publicly available.