Bridging Heterogeneous Spaces via Latent Asymmetric Bidirectional Diffusion Bridge for Human Mobility Modeling

Diffusion bridges have demonstrated remarkable success in paired translation tasks by constructing a continuous stochastic process between two coupled distributions. However, existing frameworks intrinsically rely on the assumption of homogeneous boundary domains (i.e., an assumption that requires identical topologies and dimensionalities), which renders these frameworks mathematically ill‑posed when mapping across heterogeneous state spaces. In this paper, we propose the Latent Asymmetric Bidirectional Diffusion Bridge (LAB-Bridge), a novel theoretical framework designed to construct a reversible Markov process across topologically mismatched spaces. By projecting disparate modalities into a harmonized latent Riemannian manifold via diffeomorphic mappings, we formally extend the Chapman-Kolmogorov Equations to heterogeneous boundaries. We provide a rigorous bi-Lipschitz bound guaranteeing the preservation of intrinsic topologies during this projection. Furthermore, we mathematically derive the continuous-time bidirectional Stochastic Differential Equations (SDEs) and prove the analytical equivalence of their drift fields. This equivalence enables the derivation of a highly efficient, unified asymmetric score estimator. We establish a theoretical bound demonstrating that our single-network masking paradigm optimally minimizes the Kullback-Leibler divergence of path measures in both directions simultaneously. We empirically validate LAB-Bridge on complex cross-manifold translation tasks. Specifically, we abstract human mobility dynamics as a heterogeneous state-to-state mapping challenge, demonstrating that our framework effectively translates between discrete historical preference sequences and continuous point-of-interest spatial distributions, achieving state-of-the-art bidirectional generative fidelity.