DDIM Redux: Mathematical Foundation and Some Extension

This note provides a critical review of the mathematical concepts underlying the generalized diffusion denoising implicit model (gDDIM) and the exponential integrator (EI) scheme. We present enhanced mathematical results, including an exact expression for the reverse trajectory in the probability flow ODE and an exact expression for the covariance matrix in the gDDIM scheme. Furthermore, we offer an improved understanding of the EI scheme's efficiency in terms of the change of variables. The noising process in DDIM is analyzed from the perspective of non-equilibrium statistical physics. Additionally, we propose a new scheme for DDIM, called the principal-axis DDIM (paDDIM).