A Decomposition Strategy for Probability Flow Ordinary Differential Equations in Diffusion Generative Models with Dimensionally Sharp Convergence Guarantees

Abstract

Background: Diffusion generative models have achieved remarkable success in image synthesis, audio generation, and molecular design, yet their deployment is constrained by the high computational cost of hundreds to thousands of sequential sampling steps. Existing accelerated samplers exhibit unfavorable dimensional dependence in their convergence guarantees, limiting their theoretical justification in high-dimensional practical settings. Objective: This study aims to develop a decomposition-based deterministic sampling framework for probability flow ordinary differential equations (PF-ODEs) that achieves dimensionally sharp convergence guarantees while maintaining computational efficiency. Methods: The PF-ODE is systematically partitioned into a linear variance-preserving subsystem and a nonlinear score-dependent subsystem. Sequential composition of their flow maps via a symmetric second-order Strang decomposition yields a training-free integrator. Theoretical analysis employs Baker-Campbell-Hausdorff expansions, renormalization arguments for transport equations, and stability estimates under simultaneous perturbations. Results: A non-asymptotic total variation bound TV(q̃ₕ, q) ≤ C(dε_Jac + √d ε_score + d(1 + 2√(log T))/T²) is established, reducing dimensional dependence from O(d⁶/T²) or O(d⁴/T²) of prior works to O(d/T²). Empirical validation confirms quadratic convergence (slope −1.98) on a synthetic Gaussian benchmark. Comparative experiments on CIFAR-10, CelebA, LSUN, and ImageNet subsets show superior FID against DPM-Solver, UniPC, and SA-Solver without additional runtime or memory overhead. Implications: Decomposition-based integration provides a theoretically principled and practically viable approach to accelerating diffusion sampling, bridging the gap between rigorous convergence guarantees and large-scale generative modeling applications.