Extend distributional cycle-consistency analysis beyond the Ornstein–Uhlenbeck (OU) process

Extend the distributional cycle consistency analysis for dual diffusion bridges by replacing the Ornstein–Uhlenbeck process dx_t = −x_t dt + √2 dw_t used in Appendix A-1 with a more general diffusion process, and derive the corresponding probability flow ODEs and total variation bounds that account for diffusion model training error and Runge–Kutta discretization error.

Background

The paper’s theoretical analysis in Appendix A-1 formulates the probability flow ODE and subsequent bounds under the Ornstein–Uhlenbeck (OU) process. This choice enables the authors to prove a distributional cycle-consistency bound (Theorem 1') that depends on diffusion model training error and numerical solver discretization error.

A footnote in Appendix A-1 explicitly states that the statement and argument may be extended to more general diffusion processes, but this extension is left for future work. Establishing such a generalization would broaden the theoretical foundation of dual diffusion bridges beyond the OU setting and potentially align more closely with practical diffusion processes used in generative modeling.