  Diffusion or score-based models recently showed high performance in image
generation. They rely on a forward and a backward stochastic differential
equations (SDE). The sampling of a data distribution is achieved by solving
numerically the backward SDE or its associated flow ODE. Studying the
convergence of these models necessitates to control four different types of
error: the initialization error, the truncation error, the discretization and
the score approximation. In this paper, we study theoretically the behavior of
diffusion models and their numerical implementation when the data distribution
is Gaussian. In this restricted framework where the score function is a linear
operator, we can derive the analytical solutions of the forward and backward
SDEs as well as the associated flow ODE. This provides exact expressions for
various Wasserstein errors which enable us to compare the influence of each
error type for any sampling scheme, thus allowing to monitor convergence
directly in the data space instead of relying on Inception features. Our
experiments show that the recommended numerical schemes from the diffusion
models literature are also the best sampling schemes for Gaussian
distributions.
