Convergence Theory of Decentralized Diffusion Models via Joint-Space Gaussian Cancellation

Diffusion probabilistic models (DPMs) have demonstrated success in generative modeling, motivating a growing theory of their convergence. Recent decentralized DPMs combine experts trained on separate data partitions through a learned router, raising an additional question about how routing affects sampling accuracy. We develop Joint-Space Gaussian Cancellation (JGC), a flexible convergence analysis pipeline for standard and decentralized DPMs with DDPM sampling. Our key insight is to analyze the Gaussian reverse bridge jointly with its noisy input, retaining posterior cancellations before bounding the error. An exact Gaussian-channel interpolation expresses the complete finite-step defect through a signed divergence; a posterior-covariance potential then controls its accumulation without losing an extra dimension factor. This yields an explicit $O(\{d{\log }^{2}N\}/N)$ total-variation discretization bound on equal log-SNR grids with polynomially moving SNR endpoints, merely under a finite second data moment. Here $d$ is the data dimension and $N$ the number of denoising steps. Extending the analysis classwise preserves this convergence order, while Gaussian overlap quantifies the additional routing error through expert separation without requiring a probability floor. These two results connect linear-dimensional convergence theory with the additional approximation cost of decentralized routing.