Convergence of the denoising diffusion probabilistic models

We theoretically analyze the original version of the denoising diffusion probabilistic models (DDPMs) presented in Ho, J., Jain, A., and Abbeel, P., Advances in Neural Information Processing Systems, 33 (2020), pp. 6840-6851. Our main theorem states that the sequence constructed by the original DDPM sampling algorithm weakly converges to a given data distribution as the number of time steps goes to infinity, under some asymptotic conditions on the parameters for the variance schedule, the $L^2$-based score estimation error, and the noise estimating function with respect to the number of time steps. In proving the theorem, we reveal that the sampling sequence can be seen as an exponential integrator type approximation of a reverse time stochastic differential equation (SDE). Moreover, we give a proper definition of the backward It\^o integral for general continuous processes and prove rigorously the reverse time representation of a given SDE with backward It\^o integral, without using the smoothness and uniqueness of the associated forward Kolmogorov equations.